Revision History for A096583
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| A096583
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Symmetric square array T(n,k)=T(k,n), read by antidiagonals, such that the n-th diagonal equals the convolution of the n-th row with the antidiagonal sums, with T(0,n)=1, for n>=0.
(history;
published version)
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#5 by Charles R Greathouse IV at Tue Jun 13 22:10:17 EDT 2017
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#4 by Charles R Greathouse IV at Tue Jun 13 22:09:13 EDT 2017
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| PROG
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(PARI) {) T(n, k)=if(n<0| || k<0, 0, if(n==0| || k==0, 1, if(n>k, sum(j=0, k, T(n-k, j)*sum(i=0, k-j, T(k-j-i, i))), sum(j=0, n, T(k-n, j)*sum(i=0, n-j, T(n-j-i, i))); ); ))}))); ); ))
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| STATUS
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approved
editing
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#3 by Russ Cox at Fri Mar 30 18:36:40 EDT 2012
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| AUTHOR
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_Paul D. Hanna (pauldhanna(AT)juno.com), _, Jun 28 2004
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Discussion
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| Fri Mar 30
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| OEIS Server: https://oeis.org/edit/global/213
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#2 by N. J. A. Sloane at Sat Nov 10 03:00:00 EST 2007
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| KEYWORD
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nonn,tabl,new
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| AUTHOR
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Paul D . Hanna (pauldhanna(AT)juno.com), Jun 28 2004
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#1 by N. J. A. Sloane at Wed Sep 22 03:00:00 EDT 2004
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| NAME
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Symmetric square array T(n,k)=T(k,n), read by antidiagonals, such that the n-th diagonal equals the convolution of the n-th row with the antidiagonal sums, with T(0,n)=1, for n>=0.
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| DATA
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1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 8, 7, 1, 1, 9, 16, 16, 9, 1, 1, 11, 23, 20, 23, 11, 1, 1, 13, 35, 44, 44, 35, 13, 1, 1, 15, 46, 69, 44, 69, 46, 15, 1, 1, 17, 62, 99, 108, 108, 99, 62, 17, 1, 1, 19, 77, 147, 179, 96, 179, 147, 77, 19, 1, 1, 21, 97, 206, 272, 248, 248, 272, 206, 97, 21
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| OFFSET
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0,5
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| COMMENTS
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The main diagonal (A096585) equals the partial sums of the antidiagonal sums (A096584).
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| EXAMPLE
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Antidiagonal sums are A096584 =
[1,2,5,12,24,52,90,186,306,574,...];
convolution of antidiagonal sums and first row yields main diagonal:
A096585 = [1,3,8,20,44,96,186,372,678,...];
convolution of antidiagonal sums and second row yields secondary
diagonal:
[1,5,16,44,108,248,530,1088,2138,4068,...].
Rows begin:
[1,1,1,1,1,1,1,1,1,...],
[1,3,5,7,9,11,13,15,17,...],
[1,5,8,16,23,35,46,62,77,...],
[1,7,16,20,44,69,99,147,206,...],
[1,9,23,44,44,108,179,272,379,...],
[1,11,35,69,108,96,248,429,669,...],
[1,13,46,99,179,248,186,530,965,...],
[1,15,62,147,272,429,530,372,1088,...],
[1,17,77,206,379,669,965,1088,678,...],...
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| PROG
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(PARI) {T(n, k)=if(n<0|k<0, 0, if(n==0|k==0, 1, if(n>k, sum(j=0, k, T(n-k, j)*sum(i=0, k-j, T(k-j-i, i))), sum(j=0, n, T(k-n, j)*sum(i=0, n-j, T(n-j-i, i))); ); ))}
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| CROSSREFS
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Cf. A096584, A096585.
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| KEYWORD
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nonn,tabl
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| AUTHOR
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Paul D Hanna (pauldhanna(AT)juno.com), Jun 28 2004
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| STATUS
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approved
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