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 A139377 #10 Mar 05 2018 04:55:55
%S A139377 1,1,1,3,2,1,5,6,3,1,11,15,10,4,1,21,41,30,15,5,1,43,113,92,51,21,6,1,
%T A139377 85,327,284,171,79,28,7,1,171,982,897,570,286,115,36,8,1,341,3066,
%U A139377 2895,1913,1016,446,160,45,9,1
%N A139377 A Jacobsthal-Catalan triangle.
%C A139377 First column is A001045(n+1). Second column is A139379. Row sums are A139379(n+1).
%C A139377 Diagonal sums are A135582. Inverse of the Riordan array (1-x-x^2+4x^3-2x^4,x(1-x)).
%H A139377 P. Bala, <a href="/A264772/a264772_1.pdf">A 4-parameter family of embedded Riordan arrays</a>
%F A139377 Riordan array (1/(1-x-2x^2), xc(x)) where c(x) is the g.f. of A000108
%F A139377 From _Peter Bala_, Feb 20 2018: (Start)
%F A139377 Define a(n) = floor(2^(n+2)/3) - floor(2^(n+1)/3) = A001045(n+1). Then T(n,0) = a(n) and T(n,k) = Sum_{j = 0..n-k} a(j)*k/(2*n-k-2*j)*binomial(2*n-k-2*j,n-k-j) for 1 <= k <= n.
%F A139377 Define b(n) = (2/3)*(1+i)^(n-1) + (2/3)*(1-i)^(n-1) - (4/3)*(1+i)^(n-2) - (4/3)*(1-i)^(n-2) + (1/3)*(-1)^n*Fibonacci(n+1) + (2/3)*(-1)^n*Fibonacci(n). Then T(n,k) = Sum_{j = 0..n-k} b(j)*binomial(2*n-k-j,n) for 0 <= k <= n.
%F A139377 The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the rational function (1 - 2*x)/((1 - x)*(1 + x - x^2)*(1 - 2*x + 2*x^2)) * 1/(1 - x)^n about 0. For example, for n = 4, (1 - 2*x)/((1 - x)*(1 + x - x^2)*(1 - 2*x + 2*x^2)) * 1/(1 - x)^4 = (11*x^4 + 15*x^3 + 10*x^2 + 4*x + 1) + O(x^5). (End)
%e A139377 Triangle begins
%e A139377 1;
%e A139377 1, 1;
%e A139377 3, 2, 1;
%e A139377 5, 6, 3, 1;
%e A139377 11, 15, 10, 4, 1;
%e A139377 21, 41, 30, 15, 5, 1;
%e A139377 43, 113, 92, 51, 21, 6, 1;
%e A139377 85, 327, 284, 171, 79, 28, 7, 1;
%e A139377 171, 982, 897, 570, 286, 115, 36, 8, 1;
%e A139377 The production matrix for this array is
%e A139377 1, 1,
%e A139377 2, 1, 1,
%e A139377 -2, 1, 1, 1,
%e A139377 0, 1, 1, 1, 1,
%e A139377 0, 1, 1, 1, 1, 1,
%e A139377 0, 1, 1, 1, 1, 1, 1,
%e A139377 0, 1, 1, 1, 1, 1, 1
%p A139377 #define auxiliary sequence
%p A139377 with(combinat):
%p A139377 b := proc (n)
%p A139377 (2/3)*(1+I)^(n-1) + (2/3)*(1-I)^(n-1) - (4/3)*(1+I)^(n-2)-(4/3)*(1-I)^(n-2) + (1/3)*(-1)^n*fibonacci(n+1) + (2/3)*(-1)^n*fibonacci(n);
%p A139377 end proc:
%p A139377 A139377 := proc (n, k)
%p A139377 add(b(j)*binomial(2*n-k-j, n), j = 0..n-k);
%p A139377 end proc:
%p A139377 #display sequence as a triangle
%p A139377 for n from 0 to 10 do
%p A139377 seq(A139377(n, k), k = 0..n);
%p A139377 end do; # _Peter Bala_, Feb 20 2018
%Y A139377 Cf. A001045 (first column), A139379 (second column and row sums), A135582 (sums along shallow diagonals).
%K A139377 easy,nonn,tabl
%O A139377 0,4
%A A139377 _Paul Barry_, Apr 15 2008