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Search: id:A055587
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| A055587 |
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Triangle with columns built from row sums of the partial row sums triangles obtained from Pascal's triangle A007318. Essentially A049600 formatted differently. |
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+0
9
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| 1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 8, 4, 1, 1, 16, 20, 13, 5, 1, 1, 32, 48, 38, 19, 6, 1, 1, 64, 112, 104, 63, 26, 7, 1, 1, 128, 256, 272, 192, 96, 34, 8, 1, 1, 256, 576, 688, 552, 321, 138, 43, 9, 1, 1, 512, 1280, 1696, 1520, 1002, 501, 190, 53, 10, 1, 1, 1024, 2816, 4096
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is 1/((1-z)*(1-x*z*(1-z)/(1-2*z))).
Column m (without leading zeros) is obtained from convolution of A000012 (powers of 1) with m-fold convoluted A011782.
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FORMULA
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a(n, m)= Am(n, 0) if n >= m >= 0, and a(n, m) := 0 if n<m; i.e. first column of the lower triangular matrix Am := prs^(m)(A007318) with the lower triangular matrix A007318 (Pascal triangle) and prs^(m) is the partial row sums (prs) mapping for triangular matrices applied m times. See e.g. A055584 for m=4.
G.f. for column m: (1/(1-x))*(x*(1-x)/(1-2*x))^m, m >= 0.
T(n, k) = sum_{j=0..n-k} C(n-k, j)*C(k+j-1, k-1). - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 14 2004
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EXAMPLE
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{1}; {1,1}; {1,2,1}; {1,4,3,1};...
Fourth row polynomial (n=3): p(3,x)= 1+4*x+3*x^2+x^3
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PROGRAM
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(PARI) T(n, k)=if(n<0|k<0, 0, polcoeff(polcoeff(1/((1-z)*(1-x*z*(1-z)/(1-2*z)+z*O(z^n)+x*O(x^k))), k), n)) (from Michael Somos)
(PARI) {T(n, k)=if(k>n|n<0|k<0, 0, if(k==0|k==n, 1, sum(j=0, n-k, binomial(n-k, j)*binomial(k+j-1, k-1)); ); )} (Hanna)
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CROSSREFS
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Cf. A049600, column sequences are A000012 (powers of 1), A000079 (powers of 2), A001792, A049611, A049612, A055589, A055852-5 for m=0..9, row sums: A055588.
Adjacent sequences: A055584 A055585 A055586 this_sequence A055588 A055589 A055590
Sequence in context: A101321 A091186 A138155 this_sequence A137743 A099239 A009998
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), May 30 2000
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