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.

Previous Showing 31-39 of 39 results.

A378551 a(n) = Sum_{k=0..n} 4^k * binomial(n/2+k-1,k) * binomial(n-1,n-k).

Original entry on oeis.org

1, 2, 20, 206, 2200, 24062, 267500, 3009050, 34150000, 390265190, 4484762500, 51771831146, 599921125000, 6974108163778, 81297715937500, 949957147566086, 11123368187500000, 130487420114543110, 1533247106445312500, 18042303960492212810, 212590835968046875000
Offset: 0

Views

Author

Seiichi Manyama, Nov 30 2024

Keywords

Crossrefs

Cf. A002002, A378552.
Cf. A372109.

Programs

  • Mathematica
    a[n_]:=SeriesCoefficient[ 1/(1 - 4*x/(1-x))^(n/2),{x,0,n}]; Array[a,21,0] (* Stefano Spezia, Nov 30 2024 *)
  • PARI
    a(n) = sum(k=0, n, 4^k*binomial(n/2+k-1, k)*binomial(n-1, n-k));

Formula

a(n) = [x^n] 1/(1 - 4*x/(1-x))^(n/2).

A378552 a(n) = Sum_{k=0..n} 9^k * binomial(n/3+k-1,k) * binomial(n-1,n-k).

Original entry on oeis.org

1, 3, 51, 900, 16455, 307833, 5850000, 112445112, 2180050215, 42552000000, 835075676361, 16461248223588, 325696500000000, 6464447754891285, 128654307202482420, 2566472490000000000, 51302899404879842343, 1027391467409893403745, 20607804108000000000000
Offset: 0

Views

Author

Seiichi Manyama, Nov 30 2024

Keywords

Crossrefs

Cf. A002002, A378551.
Cf. A372110.

Programs

  • Mathematica
    a[n_]:=SeriesCoefficient[1/(1 - 9*x/(1-x))^(n/3),{x,0,n}]; Array[a,19,0] (* Stefano Spezia, Nov 30 2024 *)
  • PARI
    a(n) = sum(k=0, n, 9^k*binomial(n/3+k-1, k)*binomial(n-1, n-k));

Formula

a(n) = [x^n] 1/(1 - 9*x/(1-x))^(n/3).

A378565 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(n+k-1,n-k).

Original entry on oeis.org

1, 1, 7, 43, 271, 1746, 11425, 75615, 504799, 3392953, 22930282, 155664356, 1060710457, 7250779238, 49700101101, 341474150583, 2351032782783, 16216401440106, 112035931072915, 775163096510445, 5370301986029066, 37249469056575504, 258648802856972348
Offset: 0

Views

Author

Seiichi Manyama, Dec 01 2024

Keywords

Crossrefs

Cf. A002002, A378566, A378567.
Cf. A011270, A362087.

Programs

  • Mathematica
    Table[Sum[Binomial[n+k-1, k] * Binomial[n+k-1, 2*k-1], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Dec 01 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+k-1, k)*binomial(n+k-1, n-k));

Formula

a(n) = [x^n] 1/(1 - x/(1 - x)^2)^n.
a(n) ~ (525 - 32*210^(2/3)/(157*sqrt(105) - 1575)^(1/3) + 4*(210*(157*sqrt(105) - 1575))^(1/3))^(1/6) * ((36 + (1208682 - 28350*sqrt(105))^(1/3)/3 + (6*(7461 + 175*sqrt(105)))^(1/3))^n / (2^(2/3) * 7^(1/3) * sqrt(Pi*n) * 3^(n + 1/6) * 5^(n + 1/3))). - Vaclav Kotesovec, Dec 01 2024

A378566 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(n+2*k-1,n-k).

Original entry on oeis.org

1, 1, 9, 64, 465, 3456, 26082, 199060, 1532313, 11875015, 92528414, 724187982, 5689127886, 44834549501, 354289977750, 2806262293824, 22273793685609, 177113634045858, 1410633764438967, 11251419724586850, 89860413370562730, 718528004169570925
Offset: 0

Views

Author

Seiichi Manyama, Dec 01 2024

Keywords

Crossrefs

Cf. A002002, A378565, A378567.
Cf. A052529, A362088, A365150.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+k-1, k)*binomial(n+2*k-1, n-k));

Formula

a(n) = [x^n] 1/(1 - x/(1 - x)^3)^n.

A378611 a(n) = Sum_{k=0..n} binomial(2*n+k-1,k) * binomial(n-1,n-k).

Original entry on oeis.org

1, 2, 14, 104, 806, 6412, 51908, 425476, 3520070, 29332940, 245841284, 2070093632, 17499188924, 148414157816, 1262280506144, 10762045739644, 91951462167110, 787113739061260, 6749009521216052, 57954807274992208, 498334047795436276, 4290199618047230824
Offset: 0

Views

Author

Seiichi Manyama, Dec 01 2024

Keywords

Crossrefs

Cf. A002002, A378612, A378613.
Cf. A211789, A259554.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(2*n+k-1, k)*binomial(n-1, n-k));

Formula

a(n) = [x^n] 1/(1 - x/(1 - x))^(2*n).
a(n) = (1/2)^n * [x^(2*n)] 2/(1 - x/(1 - x))^n for n > 0.
a(n) = 2 * A259554(n) for n > 0.

A341266 a(n) is the n-th term of the n-fold self-convolution of the twice left-shifted tribonacci sequence (A000073).

Original entry on oeis.org

1, 1, 5, 25, 125, 646, 3395, 18054, 96885, 523600, 2845700, 15537457, 85160387, 468279280, 2582140370, 14272523740, 79056303957, 438711518556, 2438587839980, 13574970187300, 75668677723100, 422294150816010, 2359326605275755, 13194525668986350, 73857744668632275
Offset: 0

Views

Author

Alois P. Heinz, Feb 07 2021

Keywords

Comments

The twice left-shifted tribonacci sequence begins: 1, 1, 2, 4, 7, 13, 24, ... .

Links

Crossrefs

Cf. A000073, A002002, A213684.

Programs

  • Maple
    a:= n-> coeff(series((1/(1-x-x^2-x^3))^n, x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
g:= proc(n) g(n):= `if`(n<2, (n+1)*(2-n)/2, add(g(n-j), j=1..3)) end:
b:= proc(n, k) option remember; `if`(k<2, g(n),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);

Formula

a(n) = [x^n] (1/(1-x-x^2-x^3))^n.

A378462 a(n) = Sum_{k=0..floor(n/2)} binomial(n+k-1,k) * binomial(2*n+k-1,n-2*k).

Original entry on oeis.org

1, 1, 5, 28, 157, 891, 5126, 29814, 174869, 1032481, 6128795, 36541220, 218672950, 1312712519, 7901609196, 47673716238, 288226881669, 1745734656930, 10590673033931, 64342403492274, 391414638274987, 2383907483199039, 14534764399148966, 88705912126094358
Offset: 0

Views

Author

Seiichi Manyama, Nov 27 2024

Keywords

Crossrefs

Cf. A002002, A213684.
Cf. A378467.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(n+k-1, k)*binomial(2*n+k-1, n-2*k));

Formula

a(n) = [x^n] 1/(1 - x - x^2/(1 - x)^2)^n.

A272865 Triangle read by rows, T(n,k) are covariances of inverse power traces of complex Wishart matrices with parameter c=2, for n>=1 and 1<=k<=n.

Original entry on oeis.org

4, 24, 160, 132, 936, 5700, 720, 5312, 33264, 198144, 3940, 29880, 190980, 1155600, 6823620, 21672, 167712, 1088856, 6670656, 39786120, 233908896, 119812, 941640, 6189540, 38300976, 230340740, 1363667256, 7997325700
Offset: 1

Views

Author

Fabio Deelan Cunden, May 08 2016

Keywords

Comments

These numbers provide the covariances of power traces of the time-delay matrix when the scattering matrix belongs to the Dyson ensembles.
Relation with A047781 and A002002. See eq. (60) and (61) in Cunden et al., J. Phys. A: Math. Theor. 49, 18LT01 (2016).

Examples

			Triangle starts:
4;
24,   160;
132,  936,   5700;
720,  5312,  33264,  198144;
3940, 29880, 190980, 1155600, 6823620;

References

  • F. D. Cunden, "Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities", Phys. Rev. E 91, 060102(R) (2015).
  • F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, "Correlators for the Wigner-Smith time-delay matrix of chaotic cavities", J. Phys. A: Math. Theor. 49, 18LT01 (2016).
  • F. D. Cunden, F. Mezzadri, N. O'Connell and N. Simm, "Moments of Random Matrices and Hypergeometric Orthogonal Polynomials", Commun. Math. Phys. 369, 1091-1145 (2019).

Links

Crossrefs

Cf. A002002, A047781, A050151.

Programs

  • Maple
    P := (n,k) -> simplify(n*hypergeom([1-k,k+1],[1],-1)*hypergeom([1-n,n+1],[2],-1)): seq(seq(4*(n*k)*(P(n,k)+P(k,n))/(n+k),k=1..n),n=1..7); # Peter Luschny, May 08 2016
  • Mathematica
    Clear["Global`*"];(*Wigner-Smith Covariance*)
P[k_] := Sum[Binomial[k - 1, j] Binomial[k + j, j], {j, 0, k - 1}]
Q[k_] := Sum[Binomial[k, j + 1] Binomial[k + j, j], {j, 0, k - 1}]
a[k1_, k2_] := 4 (k1 k2)/(k1 + k2) (P[k1] Q[k2] + P[k2] Q[k1])
L = 10; Table[a[k, l], {k, 1, L}, {l, 1, k}]

Formula

G.f.: ((x*y)/(x-y)^2)*((x*y-3(x+y)+1)/(sqrt(x^2-6x+1)*sqrt(y^2-6y+1))-1).
T(n,1)/4 = A050151(n) for n>=1. - Peter Luschny, May 08 2016

A350519 a(n) = A(n,n) where A(1,n) = A(n,1) = prime(n+1) and A(m,n) = A(m-1,n) + A(m,n-1) + A(m-1,n-1) for m > 1 and n > 1.

Original entry on oeis.org

3, 13, 63, 325, 1719, 9237, 50199, 275149, 1518263, 8422961, 46935819, 262512929, 1472854451, 8285893713, 46723439019, 264009961733, 1494486641911, 8473508472009, 48112827862527, 273541139290857, 1557023508876891, 8872219429659729, 50605041681538595, 288897992799897481
Offset: 1

Views

Author

Yigit Oktar, Jan 02 2022

Keywords

Comments

Replacing prime(n+1) by other functions f(n) we can get many other sequences. For example, with f(n) = 1 we get A001850.

Examples

			The two-dimensional recurrence A(m,n) can be depicted in matrix form as
   3   5   7   11   13    17    19 ...
   5  13  25   43   67    97   133 ...
   7  25  63  131  241   405   635 ...
  11  43 131  325  697  1343  2383 ...
  13  67 241  697 1719  3759  7485 ...
  17  97 405 1343 3759  9237 20481 ...
  19 133 635 2383 7485 20481 50199 ...
  ...
and then a(n) is the main diagonal of this matrix, A(n,n).

Crossrefs

Cf. A000040, A001850, A002002, A050151, A344576 (see comments).

Programs

  • MATLAB
    clear all
close all
sz = 14
f = zeros(sz,sz);
pp = primes(50);
f(1,:) = pp(2:end);
f(:,1) = pp(2:end);
for m=2:sz
    for  n=2:sz
        f(m,n) = f(m-1,n-1)+f(m,n-1)+f(m-1,n);
    end
end
an = []
for n=1:sz
    an = [an f(n,n)];
end
S = sprintf('%i,',an);
S = S(1:end-1)
  • Mathematica
    f[1,1]=3;f[m_,1]:=Prime[m+1];f[1,n_]:=Prime[n+1];f[m_,n_]:=f[m,n]=f[m-1,n]+f[m,n-1]+f[m-1,n-1];Table[f[n,n],{n,25}] (* Giorgos Kalogeropoulos, Jan 03 2022 *)
Previous Showing 31-39 of 39 results.

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