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A109692 Triangle of coefficients in expansion of (1+x)(1+3x)(1+5x)(1+7x)...(1+(2n-1)x). 8
1, 1, 1, 1, 4, 3, 1, 9, 23, 15, 1, 16, 86, 176, 105, 1, 25, 230, 950, 1689, 945, 1, 36, 505, 3480, 12139, 19524, 10395, 1, 49, 973, 10045, 57379, 177331, 264207, 135135, 1, 64, 1708, 24640, 208054, 1038016, 2924172, 4098240, 2027025, 1, 81, 2796, 53676 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 2, 3, 4, 5, 6, 7, 8, 9, ...] where DELTA is the operator defined in A084938.

LINKS

Table of n, a(n) for n=0..48.

FORMULA

a(n, m) = a(n-1,m) + (2*n-1)*a(n-1,m-1) with a(n, n) = (2*n-1)!! and a(n, 0) = 1. [Johannes W. Meijer, Jun 08 2009]

EXAMPLE

1; 1, 1; 1, 4, 3; 1, 9, 23, 15; 1, 16, 86, 176, 105; ...

MAPLE

nmax:=8; mmax:=nmax: for n from 0 to nmax do a(n, n) := doublefactorial(2*n-1) od: for n from 0 to nmax do a(n, 0):=1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := a(n-1, m) + (2*n-1)*a(n-1, m-1) od; od: seq(seq(a(n, m), m=0..n), n=0..nmax); # Johannes W. Meijer, Jun 08 2009, revised Nov 25 2012

CROSSREFS

Cf. A039758 (signed version). A028338 transposed.

Diagonals: A000012, A000290, A024196, A024197, A024198; A001147, A004041, A028339, A028340, A028341. Row sums: A000165. Central terms: A293318.

Cf. A161198 (transposed scaled triangle version).

Sequence in context: A193011 A214859 A123160 * A039758 A157894 A172106

Adjacent sequences:  A109689 A109690 A109691 * A109693 A109694 A109695

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Aug 08 2005

STATUS

approved

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Last modified August 13 06:22 EDT 2020. Contains 336442 sequences. (Running on oeis4.)