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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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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
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refs;
listen;
history;
text;
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OFFSET
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0,5
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COMMENTS
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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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LINKS
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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, Jan 14 2004
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EXAMPLE
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{1}; {1, 1}; {1, 2, 1}; {1, 4, 3, 1}; {1, 8, 8, 4, 1}; ...
Fourth row polynomial (n=3): p(3,x)= 1+4*x+3*x^2+x^3
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MATHEMATICA
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t[n_, k_] := Hypergeometric2F1[k, k-n, 1, -1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 05 2014, after Paul D. Hanna *)
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PROG
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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))}; /* Michael Somos, Sep 30 2003 */
(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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KEYWORD
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AUTHOR
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STATUS
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approved
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