diff --git a/motzkin/recurrences.py b/motzkin/recurrences.py
new file mode 100644
index 0000000..a3a79cb
--- /dev/null
+++ b/motzkin/recurrences.py
@@ -0,0 +1,115 @@
+# In this file we implement the recurrences that prove the forms of the generating function.
+import logging
+
+import logzero
+from sympy import Expr, Number, ratsimp, sqrt, var
+
+from motzkin import MotzkinPath, MotzkinSpecificationFinder
+
+logzero.loglevel(logging.WARNING)
+
+x = var("x")
+y = var("y")
+
+C = var("C")  # 2 / (1 + sqrt(1 - 4 * x ** 2))  # catalan generating function
+Cgenf = 2 / (1 + sqrt(1 - 4 * x ** 2))  # (1 - sqrt(1 - 4 * x ** 2)) / (2 * x ** 2)
+
+
+def gamma(q: MotzkinPath) -> Expr:
+    if len(q) == 0:
+        return Number(1)
+    qprime = MotzkinPath(q[:-1], pattern=True)
+    gammaprime = gamma(qprime)
+    if q[-1] == "U":
+        return ratsimp(
+            (x * y / ((1 - x) * (x - y * (1 - x))))
+            * (x * gammaprime.subs({y: x / (1 - x)}) - y * (1 - x) * gammaprime)
+        )
+    if q[-1] == "H":
+        return ratsimp(
+            # (x / (y - x - x * y ** 2))
+            (2 * x / ((y - x * C) * (1 - 2 * x * y + 1 - C * 2 * x ** 2)))
+            * (y * gammaprime - x * C * gammaprime.subs({y: x * C}))
+        )
+    if q[-1] == "D":
+        return ratsimp(
+            (x / y) * (gammaprime / (1 - x * y - x) - gammaprime.subs({y: 0}) / (1 - x))
+        )
+    raise ValueError("this isn't a motzkin path")
+
+
+def delta(q: MotzkinPath) -> Expr:
+    if len(q) == 0:
+        return Number(0)
+    qprime = MotzkinPath(q[:-1], pattern=True)
+    gammaprime = gamma(qprime)
+    deltaprime = delta(qprime)
+    if q[-1] == "D":
+        return ratsimp(deltaprime + gammaprime.subs({y: 0}) / (1 - x))
+    if q[-1] == "H":
+        return ratsimp(deltaprime + C * gammaprime.subs({y: x * C}))
+    if q[-1] == "U":
+        return ratsimp(deltaprime + gammaprime.subs({y: x / (1 - x)}) / (1 - x))
+    raise ValueError("this isn't a motzkin path")
+
+
+if __name__ == "__main__":
+    N = 10
+    # ALL:
+    msf = MotzkinSpecificationFinder([])
+    spec = msf.auto_search()
+    for i in range(5, 6):
+        for q in spec.generate_objects_of_size(i):
+            # print(q, type(q))
+            # if sum(1 for l in q if l == "H") == 3:
+            #     continue
+            # if q in (
+            #     MotzkinPath("HHHHH"),
+            #     MotzkinPath("HHHUD"),
+            #     MotzkinPath("HHUDH"),
+            #     # MotzkinPath("HUDHH"),
+            #     # MotzkinPath("HHUHD"),
+            #     # MotzkinPath("HUDUD"),
+            #     # MotzkinPath("HUHDH"),
+            #     # MotzkinPath("HUHHD"),
+            #     # MotzkinPath("HUHHD"),
+            #     # MotzkinPath("HUHHD"),
+            #     # MotzkinPath("HUUDD"),
+            #     # MotzkinPath("UDHHH"),
+            #     # MotzkinPath("UDHUD"),
+            #     # MotzkinPath("UDUDH"),
+            #     # MotzkinPath("UDUHD"),
+            #     # MotzkinPath("UHDHH"),
+            #     # MotzkinPath("UHDUD"),
+            #     # MotzkinPath("UHHDH"),
+            #     # MotzkinPath("UUDDH"),
+            #     # MotzkinPath("UHHDH"),
+            # ):
+            #     continue
+            print("Now cheking", q)
+            gf = delta(MotzkinPath(q, pattern=True))
+            actualgf = gf.subs({C: Cgenf})
+            term = actualgf.series(n=None)
+            try:
+                terms = [next(term) / x ** i for i in range(N)]
+            except:
+                print(gf)
+                print(actualgf)
+                print("Skipped", q)
+                continue
+
+            # q = "H"
+            msf = MotzkinSpecificationFinder([q])
+            qspec = msf.auto_search()
+            actual = [qspec.count_objects_of_size(i) for i in range(N)]
+
+            # print(spec.get_genf())
+            # print(gf)
+            # print(actualgf)
+
+            if actual != terms:
+                print("===== FAIL ====", q, "==== FAIL ====")
+                print(gf)
+                print(actualgf)
+                print("SPEC:", actual)
+                print("RECS:", terms)