cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A042941 Convolution of Catalan numbers A000108 with A038845.

Original entry on oeis.org

1, 13, 110, 765, 4746, 27314, 149052, 781725, 3975730, 19730150, 95973956, 459145778, 2165937060, 10095323460, 46566906872, 212857023069, 965208806082, 4345780250270, 19442667426420, 86489687956518
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Also convolution of A018218(n+1), n >= 0, with A000302 (powers of 4); also convolution of A000346 with A002697.

Links

Programs

  • Mathematica
    CoefficientList[Series[(1-Sqrt[1-4*x])/(2*x*(1-4*x)^3), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 16 2014 *)

Formula

a(n) = binomial(n+3, 2)*(4^(n+1) - A000984(n+3)/A000984(2)) / 2.
G.f.: c(x)/(1-4*x)^3, where c(x) is the g.f. for Catalan numbers.
Recurrence: (n+1)*a(n) = 128*(1-2*n)*a(n-4) + 32*(8*n-1)*a(n-3) - 24*(4*n+1)*a(n-2) + 2*(8*n+5)*a(n-1). - Fung Lam, Apr 13 2014
a(n) ~ 2^(2*n)*(n^2 - 8*n^(3/2)/(3*sqrt(Pi))). - Fung Lam, Apr 13 2014
Recurrence: n*(n+1)*a(n) = 2*n*(4*n+9)*a(n-1) - 8*(n+2)*(2*n+3)*a(n-2). - Vaclav Kotesovec, Apr 16 2014

A046527 A triangle related to A000108 (Catalan) and A000302 (powers of 4).

Original entry on oeis.org

1, 1, 1, 2, 5, 1, 5, 22, 9, 1, 14, 93, 58, 13, 1, 42, 386, 325, 110, 17, 1, 132, 1586, 1686, 765, 178, 21, 1, 429, 6476, 8330, 4746, 1477, 262, 25, 1, 1430, 26333, 39796, 27314, 10654, 2525, 362, 29, 1, 4862, 106762, 185517, 149052, 69930, 20754, 3973, 478, 33, 1
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Examples

			Triangle begins as:
     1;
     1,      1;
     2,      5,      1;
     5,     22,      9,      1;
    14,     93,     58,     13,     1;
    42,    386,    325,    110,    17,     1;
   132,   1586,   1686,    765,   178,    21,    1;
   429,   6476,   8330,   4746,  1477,   262,   25,   1;
  1430,  26333,  39796,  27314, 10654,  2525,  362,  29,  1;
  4862, 106762, 185517, 149052, 69930, 20754, 3973, 478, 33,  1;

Links

Crossrefs

Column sequences are: A000108 (k=0), A000346 (k=1), A018218 (k=2), A042941 (k=3), A042985 (k=4), A045505 (k=5), A045622 (k=6).
Row sums: A046814.

Programs

  • Magma
    A046527:= func< n,k | k eq 0 select Catalan(n) else (1/2)*Binomial(n, k-1)*(4^(n-k+1) - Binomial(2*n, n)/(k*Catalan(k-1))) >;
[A046527(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 28 2024
  • Mathematica
    T[n_, k_]:= If[k==0, CatalanNumber[n], (1/2)*Binomial[n,k-1]*(4^(n-k+ 1) -Binomial[2*n,n]/Binomial[2*(k-1),k-1])];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 28 2024 *)
  • SageMath
    def A046527(n,k):
    if k==0: return catalan_number(n)
    else: return (1/2)*binomial(n, k-1)*(4^(n-k+1) - binomial(2*n, n)/binomial(2*(k-1), k-1))
flatten([[A046527(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 28 2024

Formula

T(n, k) = binomial(n, k-1)*( 4^(n-k+1) - binomial(2*n, n)/binomial(2*(k-1), k-1) )/2, for n >= k >= 0, with T(n, 0) = A000108(n).
G.f. for column k: c(x)*(x/(1-4*x))^m, where c(x) = g.f. for Catalan numbers (A000108).

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

Views

Author

Paul Barry, Nov 09 2005

Keywords

Comments

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

Examples

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

Formula

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

A042940 Convolution of Catalan numbers A000108(n+1), n >= 0, with A038846.

Original entry on oeis.org

1, 18, 197, 1694, 12586, 84708, 530733, 3149542, 17910398, 98409532, 525628194, 2741723180, 14015785460, 70417793992, 348499310973, 1702076053686, 8216326834550, 39251274184780, 185770424237398, 871859230081092
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Also convolution of A018218(n+1), n >= 0, with itself; also convolution of A041001 with A000302 (powers of 4); also convolution of A041005 with A000984 (central binomial coefficients).

Formula

a(n) = binomial(n+4, 2)*((n+8)*A001700(2)*4^(n+1)-A002457(n+4)/2)/A002457(2), A001700(2)= 10, A002457(2)=30; G.f. (c(x)^2)/(1-4*x)^4, where c(x) = g.f. for Catalan numbers.

A090299 Table T(n,k), n>=0 and k>=0, read by antidiagonals : the k-th column given by the k-th polynomial K_k related to A090285.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 10, 5, 1, 14, 35, 22, 7, 1, 42, 126, 93, 38, 9, 1, 132, 462, 386, 187, 58, 11, 1, 429, 1716, 1586, 874, 325, 82, 13, 1, 1430, 6435, 6476, 3958, 1686, 515, 110, 15, 1, 4862, 24310, 26333, 17548, 8330, 2934, 765, 142, 17, 1
Offset: 0

Views

Author

Philippe Deléham, Jan 25 2004

Keywords

Comments

Read as a number triangle, this is the Riordan array (c(x),x/sqrt(1-4x)) where c(x) is the g.f. of A000108. - Paul Barry, May 16 2005

Examples

			row n=0 : 1, 1, 2, 5, 14, 42, 132, 429, ... see A000108.
row n=1 : 1, 3, 10, 35, 126, 462, 1716, 6435, ... see A001700.
row n=2 : 1, 5, 22, 93, 386, 1586, 6476, ... see A000346.
row n=3 : 1, 7, 38, 187, 874, 3958, 17548, ... see A000531.
row n=4 : 1, 9, 58, 325, 1686, 8330, 39796, ... see A018218.

Crossrefs

Other rows : A029887, A042941, A045724, A042985, A045492. Columns : A000012, A005408. Row n is the convolution of the row (n-j) with A000984, A000302, A002457, A002697 (first term omitted), A002802, A038845, A020918, A038846, A020920 for j=1, 2, ..9 respectively.

Formula

T(n, k) = K_k(n)= Sum_{j>=0} A090285(k, j)*2^j*binomial(n, j). T(n, 1) = 2*n+1. T(n, 2) = 2*A028387(n).

Extensions

Corrected by Alford Arnold, Oct 18 2006
Showing 1-5 of 5 results.

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