This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A372001 #24 Apr 23 2024 05:56:26 %S A372001 1,1,1,2,1,1,5,3,1,1,14,15,5,1,1,42,105,61,9,1,1,132,945,1385,297,17, %T A372001 1,1,429,10395,50521,24273,1585,33,1,1,1430,135135,2702765,3976209, %U A372001 485729,8865,65,1,1,4862,2027025,199360981,1145032281,372281761,10401345,50881,129,1,1 %N A372001 Array read by descending antidiagonals: A family of generalized Catalan numbers generated by a generalization of Deléham's Delta operator. %C A372001 Deléham's Delta operator is defined in A084938. It maps two sequences (a, b) to a triangle T. The given sequences are the coefficients of the linear function p = a + x*b which is the starting point of a recurrence described in A084938 and implemented in A371637. The generalization given here extends the number of input sequences to any number, mapping (a, b, c, ...) to p = a + x*b + x^2*c ... but leaves the recurrence unchanged. %C A372001 The result, as said, is a triangle that we can evaluate in two ways: Firstly, we only return the main diagonal. In this case, we created a new sequence from n given sequences. This case is implemented by the function A(n, dim) below. %C A372001 Alternatively, we return the entire triangle. But since the triangle is irregular, we convert it into a regular one by taking only every n-th term of a row. This case is handled by the function T(n, dim). For the first few triangles generated this way, see the link section. %H A372001 Peter Luschny, <a href="/A372001/a372001.txt">First few triangles generated by A372001.</a> %F A372001 A = DELTA([x -> (x + 1)^k : 0 <= k <= n]), i.e. here the input functions of the generalized Delta operator are the (shifted) power functions. The returned sequence is the main diagonal of the generated triangle. %e A372001 Array starts: %e A372001 [0] 1, 1, 2, 5, 14, 42, 132, ... %e A372001 [1] 1, 1, 3, 15, 105, 945, 10395, ... %e A372001 [2] 1, 1, 5, 61, 1385, 50521, 2702765, ... %e A372001 [3] 1, 1, 9, 297, 24273, 3976209, 1145032281, ... %e A372001 [4] 1, 1, 17, 1585, 485729, 372281761, 601378506737, ... %e A372001 [5] 1, 1, 33, 8865, 10401345, 38103228225, 352780110115425, ... %e A372001 [6] 1, 1, 65, 50881, 231455105, 4104215813761, 220579355255364545, ... %e A372001 . %e A372001 Seen as a triangle T(n, k) = A(k, n - k): %e A372001 [0] [ 1] %e A372001 [1] [ 1, 1] %e A372001 [2] [ 2, 1, 1] %e A372001 [3] [ 5, 3, 1, 1] %e A372001 [4] [ 14, 15, 5, 1, 1] %e A372001 [5] [ 42, 105, 61, 9, 1, 1] %e A372001 [6] [132, 945, 1385, 297, 17, 1, 1] %e A372001 [7] [429, 10395, 50521, 24273, 1585, 33, 1, 1] %o A372001 (SageMath) %o A372001 def GeneralizedDelehamDelta(F, dim, seq=True): # The algorithm. %o A372001 ring = PolynomialRing(ZZ, 'x') %o A372001 x = ring.gen() %o A372001 A = [sum(F[j](k) * x^j for j in range(len(F))) for k in range(dim)] %o A372001 C = [ring(0)] + [ring(1) for i in range(dim)] %o A372001 for k in range(dim): %o A372001 for n in range(k, 0, -1): %o A372001 C[n] = C[n-1] + C[n+1] * A[n-1] %o A372001 yield list(C[1])[-1] if seq else list(C[1]) %o A372001 def F(n): # Define the input functions. %o A372001 def p0(): return lambda n: pow(n, n^0) %o A372001 def p(k): return lambda n: pow(n + 1, k) %o A372001 return [p0()] + [p(k) for k in range(n + 1)] %o A372001 def A(n, dim): # Return only the main diagonal of the triangle. %o A372001 return [r for r in GeneralizedDelehamDelta(F(n), dim)] %o A372001 for n in range(7): print(A(n, 7)) %o A372001 def T(n, dim): # Return the regularized triangle. %o A372001 R = GeneralizedDelehamDelta(F(n), dim, False) %o A372001 return [[r[k] for k in range(0, len(r), n + 1)] for r in R] %o A372001 for n in range(0, 4): %o A372001 for row in T(n, 6): print(row) %Y A372001 By ascending antidiagonals: A290569. %Y A372001 Family: A000108 (n=0), A001147 (n=1), A000364 (n=2), A216966 (n=3), A227887 (n=4), A337807 (n=5), A337808 (n=6), A337809 (n=7). %Y A372001 Cf. A291333 (main diagonal), A371999 (row sums of triangle). %Y A372001 Cf. A084938, A131427, A371994, A371637. %K A372001 nonn,tabl %O A372001 1,4 %A A372001 _Peter Luschny_, Apr 21 2024