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 1-4 of 4 results.

A045505 Convolution of A000108 (Catalan numbers) with A040075.

Original entry on oeis.org

1, 21, 262, 2525, 20754, 152946, 1040556, 6659037, 40599130, 237978598, 1350216660, 7453221490, 40188242420, 212349718980, 1102352779992, 5634083759325, 28400234400810, 141402315307550, 696257439473860
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Also convolution of A045492 with A000984 (central binomial coefficients); also convolution of A042985 with A000302 (powers of 4).

Links

Programs

  • GAP
    List([0..20], n-> Binomial(n+5,4)*(2^(2*n+1) - Binomial(2*n+10,n+5)/140)); # G. C. Greubel, Jan 13 2020
  • Magma
    [Binomial(n+5,4)*(2^(2*n+1) - Binomial(2*n+10,n+5)/140): n in [0..20]]; // G. C. Greubel, Jan 13 2020
  • Maple
    seq(coeff(series((1-sqrt(1-4*x))/(2*x*(1-4*x)^5), x, n+1), x, n), n = 0..20); # G. C. Greubel, Jan 13 2020
  • Mathematica
    Table[Binomial[n+5,4]*(2^(2*n+1) -Binomial[2*n+10, n+5]/140), {n,0,20}] (* G. C. Greubel, Jan 13 2020 *)
  • PARI
    vector(21, n, binomial(n+5,4)*(2^(2*n+1) -binomial(2*n+10,n+5)/140)) \\ G. C. Greubel, Jan 13 2020
  • Sage
    [binomial(n+5,4)*(2^(2*n+1) - binomial(2*n+10,n+5)/140) for n in (0..20)] # G. C. Greubel, Jan 13 2020

Formula

a(n) = binomial(n+5, 4)*(4^(n+1) - A000984(n+5)/A000984(4))/2, A000984(n) = binomial(2*n, n).
G.f. c(x)/(1-4*x)^5, where c(x) = g.f. for Catalan numbers.

A045530 Convolution of A000108 (Catalan numbers) with A020922.

Original entry on oeis.org

1, 23, 310, 3195, 27866, 216566, 1546028, 10338515, 65635570, 399429602, 2346750900, 13384232030, 74417751940, 404759481420, 2159510136408, 11327603405955, 58528412321250, 298354368109930, 1502525977613540
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Also convolution of A045505 with A000984 (central binomial coefficients); also convolution of A045492 with A000302 (powers of 4).

Links

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (Sqrt(1-4*x) +4*x-1)/(2*x*(1-4*x)^6) )); // G. C. Greubel, Jan 13 2020
  • Maple
    seq(coeff(series((sqrt(1-4*x) +4*x-1)/(2*x*(1-4*x)^6), x, n+1), x, n), n = 0..40); # G. C. Greubel, Jan 13 2020
  • Mathematica
    CoefficientList[Series[(Sqrt[1-4*x] +4*x-1)/(2*x*(1-4*x)^6), {x,0,40}], x] (* G. C. Greubel, Jan 13 2020 *)
  • PARI
    my(x='x+O('x^40)); Vec( (sqrt(1-4*x) +4*x-1)/(2*x*(1-4*x)^6) ) \\ G. C. Greubel, Jan 13 2020
  • Sage
    def A045530_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (sqrt(1-4*x) +4*x-1)/(2*x*(1-4*x)^6) ).list()
A045530_list(40) # G. C. Greubel, Jan 13 2020

Formula

a(n) = binomial(n+6, 5)*(A000984(n+6)/A000984(5) - 5*4^(n+1)/(n+6))/2, A000984(n) = binomial(2*n, n).
G.f. c(x)/(1-4*x)^(11/2), where c(x) = g.f. for Catalan numbers.

A046658 Triangle related to A001700 and A000302 (powers of 4).

Original entry on oeis.org

1, 3, 1, 10, 7, 1, 35, 38, 11, 1, 126, 187, 82, 15, 1, 462, 874, 515, 142, 19, 1, 1716, 3958, 2934, 1083, 218, 23, 1, 6435, 17548, 15694, 7266, 1955, 310, 27, 1, 24310, 76627, 80324, 44758, 15086, 3195, 418, 31, 1, 92378, 330818, 397923, 259356, 105102, 27866, 4867, 542, 35, 1
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Examples

			Triangle begins as:
      1;
      3,     1;
     10,     7,     1;
     35,    38,    11,     1;
    126,   187,    82,    15,     1;
    462,   874,   515,   142,    19,    1;
   1716,  3958,  2934,  1083,   218,   23,   1;
   6435, 17548, 15694,  7266,  1955,  310,  27,  1;
  24310, 76627, 80324, 44758, 15086, 3195, 418, 31, 1;

Links

Crossrefs

Column sequences for m=1..6: A001700, A000531, A029887, A045724, A045492, A045530.
Row sums: A046885.
Cf. A000302.

Programs

  • Magma
    A046658:= func< n,k | Binomial(n,k)*(Binomial(n+1,2)*Catalan(n )/Catalan(k-1) -4^(n-k+1)*Binomial(k,2))/(n*(n-k+1)) >;
[A046658(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 28 2024
  • Mathematica
    T[n_, k_]:= (1/2)*Binomial[n,k-1]*(Binomial[2*n,n]/Binomial[2*(k-1), k -1] - 4^(n-k+1)*(k-1)/n);
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 28 2024 *)
  • SageMath
    def A046658(n,k): return (1/2)*binomial(n,k-1)*(binomial(2*n, n)/binomial(2*(k-1), k-1) - 4^(n-k+1)*(k-1)/n)
flatten([[A046658(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Jul 28 2024

Formula

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

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 1-4 of 4 results.

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