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

0,4

Comments

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.

Links

Indranil Ghosh, Rows 1..100, flattened

Formula

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, k>=0} T(n-2*k,k) = A001045(n+1).

Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A008346(n), A006053(n+2), A001654(n+1) for x = -1, 0, 1 respectively.

Example

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

Mathematica

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 *)

Crossrefs

Cf. A006053, A000012, A000027, A055998

Sequence in context: A271513 A306801 A117212 * A348013 A105033 A092486

Adjacent sequences: A208150 A208151 A208152 * A208154 A208155 A208156

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Feb 24 2012

Status

approved