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Revision History for A126150 (Bold, blue-underlined text is an addition; faded, red-underlined text is a deletion.)

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Symmetric triangle, read by rows of 2*n+1 terms, similar to triangle A008301.
(history; published version)
#9 by Charles R Greathouse IV at Wed Jun 14 00:22:15 EDT 2017
STATUS

editing

approved

#8 by Charles R Greathouse IV at Wed Jun 14 00:22:09 EDT 2017
PROG

(PARI) T(n, k)=local(p=3); if(2*n<k || k<0, 0, if(n==0 && k==0, 1, if(k==0, sum(j=0, 2*n-2, T(n-1, j)), if(k==1, (p+1)*T(n, 0), if(k<=n, 2*T(n, k-1)-T(n, k-2)-2*p*T(n-1, k-2), T(n, 2*n-k))))))

(PARI) /* Alternate Recurrence: */ T(n, k)=local(p=3); if(2*n<k || k<0, 0, if(n==0 && k==0, 1, if(k==0, sum(j=0, 2*n-2, T(n-1, j)), if(k<=n, T(n, k-1)+p*sum(j=k-1, 2*n-1-k, T(n-1, j)), T(n, 2*n-k)))))

STATUS

approved

editing

#7 by Charles R Greathouse IV at Sun Jun 14 11:19:49 EDT 2015
STATUS

editing

approved

#6 by Charles R Greathouse IV at Sun Jun 14 11:19:45 EDT 2015
PROG

(PARI) {T(n, k)=local(p=3); if(2*n<k|k<0, 0, if(n==0&k==0, 1, if(k==0, sum(j=0, 2*n-2, T(n-1, j)), if(k==1, (p+1)*T(n, 0), if(k<=n, 2*T(n, k-1)-T(n, k-2)-2*p*T(n-1, k-2), T(n, 2*n-k))))))} (PARI) /* Alternate Recurrence: */ {T(n, k)=local(p=3); if(2*n<k|k<0, 0, if(n==0&k==0, 1, if(k==0, sum(j=0, 2*n-2, T(n-1, j)), if(k<=n, T(n, k-1)+p*sum(j=k-1, 2*n-1-k, T(n-1, j)), T(n, 2*n-k)))))}

(PARI) /* Alternate Recurrence: */ T(n, k)=local(p=3); if(2*n<k|k<0, 0, if(n==0&k==0, 1, if(k==0, sum(j=0, 2*n-2, T(n-1, j)), if(k<=n, T(n, k-1)+p*sum(j=k-1, 2*n-1-k, T(n-1, j)), T(n, 2*n-k)))))

STATUS

approved

editing

#5 by Russ Cox at Fri Mar 30 18:37:02 EDT 2012
AUTHOR

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Dec 19 2006

Discussion
Fri Mar 30
18:37
OEIS Server: https://oeis.org/edit/global/213
#4 by N. J. A. Sloane at Sun Jun 29 03:00:00 EDT 2008
EXAMPLE

20256 = 2*11904 - 2976 - 6*96 ;

26304 = 2*20256 - 11904 - 6*384 ;

28536 = 2*26304 - 20256 - 6*636 ;

26304 = 2*28536 - 26304 - 6*744 ;

20256 = 2*26304 - 28536 - 6*636 ;

11904 = 2*20256 - 26304 - 6*384 ;

KEYWORD

nonn,tabl,new

#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), Dec 19 2006

#2 by N. J. A. Sloane at Fri May 11 03:00:00 EDT 2007
EXAMPLE

2976 = 2*11904 - 20256 - 6*96 .

KEYWORD

nonn,tabl,new

#1 by N. J. A. Sloane at Fri Jan 12 03:00:00 EST 2007
NAME

Symmetric triangle, read by rows of 2*n+1 terms, similar to triangle A008301.

DATA

1, 1, 4, 1, 6, 24, 36, 24, 6, 96, 384, 636, 744, 636, 384, 96, 2976, 11904, 20256, 26304, 28536, 26304, 20256, 11904, 2976, 151416, 605664, 1042056, 1407024, 1650456, 1736064, 1650456, 1407024, 1042056, 605664, 151416, 11449296, 45797184

OFFSET

0,3

FORMULA

Sum_{k=0,2n} (-1)^k*C(2n,k)*T(n,k) = (-6)^n.

EXAMPLE

Triangle begins:

1;

1, 4, 1;

6, 24, 36, 24, 6;

96, 384, 636, 744, 636, 384, 96;

2976, 11904, 20256, 26304, 28536, 26304, 20256, 11904, 2976;

151416, 605664, 1042056, 1407024, 1650456, 1736064, 1650456, 1407024, 1042056, 605664, 151416; ...

If we write the triangle like this:

.......................... ....1;

................... ....1, ....4, ....1;

............ ....6, ...24, ...36, ...24, ....6;

..... ...96, ..384, ..636, ..744, ..636, ..384, ...96;

.2976, 11904, 20256, 26304, 28536, 26304, 20256, 11904, .2976;

then the first term in each row is the sum of the previous row:

2976 = 96 + 384 + 636 + 744 + 636 + 384 + 96

the next term is 4 times the first:

11904 = 4*2976,

and the remaining terms in each row are obtained by the rule

illustrated by:

20256 = 2*11904 - 2976 - 6*96 ;

26304 = 2*20256 - 11904 - 6*384 ;

28536 = 2*26304 - 20256 - 6*636 ;

26304 = 2*28536 - 26304 - 6*744 ;

20256 = 2*26304 - 28536 - 6*636 ;

11904 = 2*20256 - 26304 - 6*384 ;

2976 = 2*11904 - 20256 - 6*96 .

An alternate recurrence is illustrated by:

11904 = 2976 + 3*(96 + 384 + 636 + 744 + 636 + 384 + 96);

20256 = 11904 + 3*(384 + 636 + 744 + 636 + 384);

26304 = 20256 + 3*(636 + 744 + 636);

28536 = 26304 + 3*(744);

and then for k>n, T(n,k) = T(n,2n-k).

PROG

(PARI) {T(n, k)=local(p=3); if(2*n<k|k<0, 0, if(n==0&k==0, 1, if(k==0, sum(j=0, 2*n-2, T(n-1, j)), if(k==1, (p+1)*T(n, 0), if(k<=n, 2*T(n, k-1)-T(n, k-2)-2*p*T(n-1, k-2), T(n, 2*n-k))))))} (PARI) /* Alternate Recurrence: */ {T(n, k)=local(p=3); if(2*n<k|k<0, 0, if(n==0&k==0, 1, if(k==0, sum(j=0, 2*n-2, T(n-1, j)), if(k<=n, T(n, k-1)+p*sum(j=k-1, 2*n-1-k, T(n-1, j)), T(n, 2*n-k)))))}

CROSSREFS

Cf. A126151 (column 0); diagonals: A126152, A126153; A126154; variants: A008301, A125053, A126155.

KEYWORD

nonn,tabl

AUTHOR

Paul D Hanna (pauldhanna(AT)juno.com), Dec 19 2006

STATUS

approved