A038836 -id:A038836 - cp's OEIS frontend

A041001 Convolution of A000108(n+1), n >= 0, (Catalan numbers) with A038845 (3-fold convolution of powers of 4).

1, 14, 125, 906, 5810, 34364, 191901, 1026610, 5312230, 26767940, 131990066, 639210404, 3048892740, 14354652152, 66828135005, 308078809794, 1408022619806, 6385966846580, 28765327498278, 128777533131500

Offset: 0

Wolfdieter Lang

Also convolution of A038836 with A000984 (central binomial coefficients); also convolution of A001791(n+1), n >= 0, with A002802; also convolution of A008549(n+1), n >= 0, with A002697; also convolution of A029760 with A002457; also convolution of A038806(n+1), n >= 0, with A000302 (powers of 4).

a(n) = (n+3)*(3*(n+6)*2^(2*n+3)-(n+4)*binomial(2*n+7, n+3))/12; G.f. (c(x)^2)/(1-4*x)^3, where c(x) = g.f. for Catalan numbers.

A113955 Riordan array (1/((1-4x)c(x)),xc(x)/sqrt(1-4x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 3, 1, 11, 6, 1, 42, 30, 9, 1, 163, 140, 58, 12, 1, 638, 630, 325, 95, 15, 1, 2510, 2772, 1686, 624, 141, 18, 1, 9908, 12012, 8330, 3682, 1064, 196, 21, 1, 39203, 51480, 39796, 20264, 7050, 1672, 260, 24, 1, 155382, 218790, 185517, 106203, 42849, 12303, 2475

Offset: 0

Paul Barry, Nov 09 2005

Columns include A032443,A002457,A018218,A038836. Row sums are A100192. Diagonal sums are A113956.

			Triangle begins
1;
3, 1;
11, 6, 1;
42, 30, 9, 1;
163, 140, 58, 12, 1;
638, 630, 325, 95, 15, 1;

Riordan array ((1/(1-4x)+1/sqrt(1-4x))/2, (2x/((1-4x)+sqrt(1-4x)))); Number triangle T(n, k)=sum{j=0..n, C(j, j-k)C(2n, n-j)}.

T(n,k)=sum{j=0..n, C(2n,j)C(n-j,k)}; - Paul Barry, Apr 03 2006

A041005 Convolution of Catalan numbers A000108(n+1), n >= 0, with A020918.

Original entry on oeis.org

1, 16, 159, 1260, 8722, 55152, 326811, 1844084, 10015566, 52754624, 270976342, 1362986520, 6734927460, 32775704608, 157408497171, 747269225028, 3511471892470, 16351481223840, 75525932249922, 346305571781224

Offset: 0

Wolfdieter Lang

Also convolution of A001791(n+1), n >= 0, with A038845; also convolution of A008549(n+1), n >= 0, with A002802; also convolution of A029760 with A002697; also convolution of A038806(n+1), n >= 0, with A002457; also convolution of A038836 with A000302 (powers of 4); also convolution of A041001 with A000984 (central binomial coefficients).

a(n)=binomial(n+7, 3)*binomial(2*(n+4), n+2)/20 - (n+4)*(n+3)*4^(n+1); G.f. (c(x)^2)/(1-4*x)^(7/2), where c(x) = g.f. for Catalan numbers.

cp's OEIS Frontend

A041001 Convolution of A000108(n+1), n >= 0, (Catalan numbers) with A038845 (3-fold convolution of powers of 4).

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A113955 Riordan array (1/((1-4x)c(x)),xc(x)/sqrt(1-4x)), c(x) the g.f. of A000108.

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Formula

A041005 Convolution of Catalan numbers A000108(n+1), n >= 0, with A020918.

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Author

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Formula