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Revision History for A113716

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Triangle, read by rows, where row n forms a polynomial in y=3*k that generates diagonal n as k=0,1,2,... for n>=0; thus T(n,k) = Sum_{j=0..n-k} T(n-k,j)*(3*k)^j, with T(n,0)=T(n,n)=1.
(history; published version)

#5 by Charles R Greathouse IV at Tue Jun 13 23:36:18 EDT 2017

STATUS

editing

approved

#4 by Charles R Greathouse IV at Tue Jun 13 23:36:14 EDT 2017

PROG

(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k || k==0, 1, sum(j=0, n-k, T(n-k, j)*(3*k)^j)))

STATUS

approved

editing

#3 by Russ Cox at Fri Mar 30 18:36:52 EDT 2012

AUTHOR

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Nov 08 2005

Discussion

Fri Mar 30

18:36

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

#2 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), Nov 08 2005

#1 by N. J. A. Sloane at Tue Jan 24 03:00:00 EST 2006

NAME

Triangle, read by rows, where row n forms a polynomial in y=3*k that generates diagonal n as k=0,1,2,... for n>=0; thus T(n,k) = Sum_{j=0..n-k} T(n-k,j)*(3*k)^j, with T(n,0)=T(n,n)=1.

DATA

1, 1, 1, 1, 4, 1, 1, 22, 7, 1, 1, 157, 61, 10, 1, 1, 1372, 601, 118, 13, 1, 1, 14008, 6595, 1495, 193, 16, 1, 1, 161995, 79981, 20206, 3001, 286, 19, 1, 1, 2079994, 1065589, 291394, 48685, 5281, 397, 22, 1, 1, 29268778, 15495415, 4492621, 825313, 100456

OFFSET

0,5

EXAMPLE

Triangle begins:

1;

1,1;

1,4,1;

1,22,7,1;

1,157,61,10,1;

1,1372,601,118,13,1;

1,14008,6595,1495,193,16,1;

1,161995,79981,20206,3001,286,19,1;

1,2079994,1065589,291394,48685,5281,397,22,1;

1,29268778,15495415,4492621,825313,100456,8497,526,25,1; ...

where diagonals are generated by row polynomials:

T(6,5) = (1) + (1)*(3*5) = 16.

T(6,4) = (1) + (4)*(3*4) + (1)*(3*4)^2 = 193.

T(6,3) = (1) + (22)*(3*3) + (7)*(3*3)^2 + (1)*(3*3)^3 = 1495.

T(6,2) = (1) + (157)*(3*2) + (61)*(3*2)^2 + (10)*(3*2)^3 + (1)*(3*2)^4 = 6595.

PROG

(PARI) T(n, k)=if(n<k|k<0, 0, if(n==k|k==0, 1, sum(j=0, n-k, T(n-k, j)*(3*k)^j)))

CROSSREFS

Cf. A091150, A113711, A113717 (column 1), A113718 (column 2), A113719 (row sums).

KEYWORD

nonn,tabl

AUTHOR

Paul D Hanna (pauldhanna(AT)juno.com), Nov 08 2005

STATUS

approved