The On-Line Encyclopedia of Integer Sequences (OEIS)

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A088925

Square table, read by antidiagonals, of coefficients T(n,k) of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xy*f(x,y)^3.
(history; published version)

#9 by Bruno Berselli at Mon Nov 18 10:03:46 EST 2013

STATUS

proposed

approved

#8 by Michel Marcus at Mon Nov 18 09:43:45 EST 2013

STATUS

editing

proposed

#7 by Michel Marcus at Mon Nov 18 09:43:18 EST 2013

FORMULA

T(n, k) = sum(i=0, k, C(n+k, 2i)*C(n+k-2i, k-i)*A001764(i) ), where A001764(i)=(3i)!/[(i!(2i+1)! ] (). - from _Michael Somos)._

STATUS

proposed

editing

#6 by Jean-François Alcover at Mon Nov 18 09:19:57 EST 2013

STATUS

editing

proposed

#5 by Jean-François Alcover at Mon Nov 18 09:19:51 EST 2013

MATHEMATICA

t[n_, k_] := Sum[ Binomial[n+k, 2*i]*Binomial[n+k-2*i, k-i]*(3*i)!/(i!*(2*i+1)!), {i, 0, k}]; Table[t[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 18 2013, after Michael Somos *)

STATUS

approved

editing

#4 by Russ Cox at Fri Mar 30 18:36:39 EDT 2012

AUTHOR

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Oct 23 2003

Discussion

Fri Mar 30

18:36

OEIS Server: https://oeis.org/edit/global/213

#3 by N. J. A. Sloane at Sat Nov 10 03:00:00 EST 2007

KEYWORD

nonn,tabl,new

AUTHOR

Paul D . Hanna (pauldhanna(AT)juno.com), Oct 23 2003

#2 by N. J. A. Sloane at Fri Feb 24 03:00:00 EST 2006

FORMULA

T(n, k) = sum(i=0, k, C(n+k, 2i)*C(n+k-2i, k-i)*A001764(i) ), where A001764(i)=(3i)!/[i!(2i+1)! ] (from Michael Somos).

KEYWORD

nonn,tabl,new

#1 by N. J. A. Sloane at Thu Feb 19 03:00:00 EST 2004

NAME

Square table, read by antidiagonals, of coefficients T(n,k) of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xy*f(x,y)^3.

DATA

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 21, 10, 1, 1, 15, 55, 55, 15, 1, 1, 21, 120, 212, 120, 21, 1, 1, 28, 231, 644, 644, 231, 28, 1, 1, 36, 406, 1652, 2617, 1652, 406, 36, 1, 1, 45, 666, 3738, 8685, 8685, 3738, 666, 45, 1, 1, 55, 1035, 7680, 24735, 36345, 24735, 7680

OFFSET

0,5

COMMENTS

The g.f. for A001764 satisfies: g(x) = 1 + x*g(x)^3.

FORMULA

T(n,k) = sum(i=0,k, C(n+k,2i)*C(n+k-2i,k-i)*A001764(i) ), where A001764(i)=(3i)!/[i!(2i+1)! ] (from Michael Somos).

EXAMPLE

Rows begin:

{1, 1, 1, 1, 1, 1, 1, 1,..}

{1, 3, 6,10,15,21,28,..}

{1, 6,21,55,120,231,..}

{1,10,55,212,644,..}

{1,15,120,644,..}

{1,21,231,..}

CROSSREFS

Cf. A088926 (diagonal), A088927 (antidiagonal sums), A086617, A001764.

KEYWORD

nonn,tabl

AUTHOR

Paul D Hanna (pauldhanna(AT)juno.com), Oct 23 2003

STATUS

approved