A126952 -id:A126952 - cp's OEIS frontend

A171243 Riordan array (f(x), x*g(x)), f(x) is the g.f. of A126952, g(x) is the g.f. of A117641.

1, 1, 1, 5, 1, 1, 21, 6, 1, 1, 93, 25, 7, 1, 1, 421, 112, 29, 8, 1, 1, 1937, 510, 132, 33, 9, 1, 1, 9017, 2357, 606, 153, 37, 10, 1, 1, 42349, 11009, 2819, 709, 175, 41, 11, 1, 1, 200277, 51840, 13233, 3324, 819, 198, 45, 12, 1, 1

Offset: 0

Philippe Deléham, Dec 06 2009

Expansion of row sums of T_(x,3), T_(x,y) defined in A039599.

Matrix product P^3 * Q * P^(-3), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793 and A158815. - Peter Bala, Jul 13 2021

			Triangle begins:
    1;
    1,   1;
    5,   1,  1;
   21,   6,  1, 1;
   93,  25,  7, 1, 1;
  421, 112, 29, 8, 1, 1;
  ...

Cf. A061554, A158793, A158815, A171224.

Sum_{k=0..n} T(n,k)*x^k = A126952(n), A126568(n), A026375(n), A026378(n+1), A000351(n) for x = 0,1,2,3,4 respectively.

A126970 Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = T(n-1,1), T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + T(n-1,k+1) for k >= 1.

Original entry on oeis.org

1, 0, 1, 1, 3, 1, 3, 11, 6, 1, 11, 42, 30, 9, 1, 42, 167, 141, 58, 12, 1, 167, 684, 648, 327, 95, 15, 1, 684, 2867, 2955, 1724, 627, 141, 18, 1, 2867, 12240, 13456, 8754, 3746, 1068, 196, 21, 1, 12240, 53043, 61362, 43464, 21060, 7146, 1677, 260, 24, 1

Offset: 0

Philippe Deléham, Mar 19 2007

This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007

			Triangle begins:
    1;
    0,   1;
    1,   3,   1;
    3,  11,   6,   1;
   11,  42,  30,   9,  1;
   42, 167, 141,  58, 12,  1;
  167, 684, 648, 327, 95, 15, 1; ...
From _Philippe Deléham_, Nov 07 2011: (Start)
Production matrix begins:
  0, 1
  1, 3, 1
  0, 1, 3, 1
  0, 0, 1, 3, 1
  0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 0, 0, 1, 3, 1 (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0,  T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]]; Table[T[n, k, 0, 3], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)

Sum_{k=0..n} T(n,k) = A126952(n).
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A117641(m+n).
Sum_{k=0..n} T(n,k)*(4*k+1) = 5^n. - Philippe Deléham, Mar 22 2007

cp's OEIS Frontend

A171243 Riordan array (f(x), x*g(x)), f(x) is the g.f. of A126952, g(x) is the g.f. of A117641.

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