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Search: id:A100324
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| A100324 |
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Square array, read by antidiagonals, where rows are successive self-convolutions of the top row, which equals A003169 shifted one place right. |
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6
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| 1, 1, 1, 1, 2, 3, 1, 3, 7, 14, 1, 4, 12, 34, 79, 1, 5, 18, 61, 195, 494, 1, 6, 25, 96, 357, 1230, 3294, 1, 7, 33, 140, 575, 2277, 8246, 22952, 1, 8, 42, 194, 860, 3716, 15372, 57668, 165127, 1, 9, 52, 259, 1224, 5641, 25298, 108018, 415995, 1217270, 1, 10, 63, 336
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Main diagonal: T(n,n) = (n+1)*A032349(n+1) for n>0 with T(0,0) = 1. Antidiagonal sums form A100325. Also, column k forms the binomial transform of row k in triangle A100326 for k>=0.
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FORMULA
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T(n, k) = Sum_{i=0..k} T(0, k-i)*T(n-1, i) for n>0. T(0, k) = A003169(k+1) = ( (324*k^2-708*k+360)*T(0, k-1) - (371*k^2-1831*k+2250)*T(0, k-2) +(20*k^2-130*k+210)*T(0, k-3) )/(16*k*(2*k-1)) for k>2, with T(0, 0)=T(0, 1)=1, T(0, 2)=3.
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EXAMPLE
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Rows begin:
[1,1,3,14,79,494,3294,...],
[1,2,7,34,195,1230,8246,...],
[1,3,12,61,357,2277,15372,...],
[1,4,18,96,575,3716,25298,...],
[1,5,25,140,860,5641,38775,...],
[1,6,33,194,1224,8160,56695,...],
[1,7,42,259,1680,11396,80108,...],...
Main diagonal = [1,2*1,3*4,4*24,5*172,6*1360,...,(n+1)*A032349(n+1),...].
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PROGRAM
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(PARI) {T(n, k)=if(k==0, 1, if(n>0, sum(i=0, k, T(0, k-i)*T(n-1, i)), if(k==1, 1, if(k==2, 3, ( (324*k^2-708*k+360)*T(0, k-1)-(371*k^2-1831*k+2250)*T(0, k-2)+(20*k^2-130*k+210)*T(0, k-3))/(16*k*(2*k-1)) ))); )}
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CROSSREFS
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Cf. A003169, A032349, A100325, A100326.
Sequence in context: A102473 A011117 A069269 this_sequence A121424 A093768 A119011
Adjacent sequences: A100321 A100322 A100323 this_sequence A100325 A100326 A100327
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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), Nov 16 2004
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