A188107 -id:A188107 - cp's OEIS frontend

A208153 Convolution triangle based on A006053.

1, 1, 1, 3, 2, 1, 4, 7, 3, 1, 9, 14, 12, 4, 1, 14, 35, 31, 18, 5, 1, 28, 70, 87, 56, 25, 6, 1, 47, 154, 207, 175, 90, 33, 7, 1, 89, 306, 504, 476, 310, 134, 42, 8, 1, 155, 633, 1137, 1274, 941, 504, 189, 52, 9, 1

Offset: 0

Philippe Deléham, Feb 24 2012

Riordan array (1/(1-x-2*x^2+x^3), x/(1-x-2*x^2+x^3)).
Subtriangle of triangle given by (0, 1, 2, -5/2, 1/10, 2/5, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Diagonal sums are A125691(n).
Row sums are A001654(n+1).
Mirror image of triangle in A188107.

			Triangle begins:
  1
  1, 1
  3, 2, 1
  4, 7, 3, 1
  9, 14, 12, 4, 1
  14, 35, 31, 18, 5, 1
Triangle (0, 1, 2, -5/2, 1/10, 2/5, 0, 0,...) DELTA (1, 0, 0, 0,...) begins:
  1
  0, 1
  0, 1, 1
  0, 3, 2, 1
  0, 4, 7, 3, 1
  0, 9, 14, 12, 4, 1
  0, 14, 35, 31, 18, 5, 1

Indranil Ghosh, Rows 1..100, flattened

Cf. A006053, A000012, A000027, A055998.

nmax=9; Flatten[CoefficientList[Series[CoefficientList[Series[1/(1 - x - 2*x^2 + x^3 - y*x), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 10 2017 *)

T(n,k) = T(n-1,k-1) + T(n-1,k) + 2*T(n-2,k) - T(n-3,k).
G.f.: 1/(1-x-2*x^2+x^3-y*x).
Sum_{k>=0} T(n-2*k,k) = A001045(n+1).
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A008346(n), A006053(n+2), A001654(n+1) for x = -1, 0, 1 respectively.

cp's OEIS Frontend

A208153 Convolution triangle based on A006053.

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