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A208153 Convolution triangle based on A006053. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A105033 A092486 A159966

Adjacent sequences:  A208150 A208151 A208152 * A208154 A208155 A208156

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Feb 24 2012

STATUS

approved

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Last modified July 29 05:47 EDT 2021. Contains 346340 sequences. (Running on oeis4.)