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.

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A045543 6-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^6.

Original entry on oeis.org

1, 24, 336, 3584, 32256, 258048, 1892352, 12976128, 84344832, 524812288, 3148873728, 18320719872, 103817412608, 574988746752, 3121367482368, 16647293239296, 87398289506304, 452414675091456, 2312341672689664, 11683410556747776, 58417052783738880, 289303499500421120
Offset: 0

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Author

Wolfdieter Lang

Keywords

Comments

Also convolution of A020922 with A000984 (central binomial coefficients); also convolution of A040075 with A000302 (powers of 4).
With a different offset, number of n-permutations of 5 objects: u,v,z,x, y with repetition allowed, containing exactly five (5) u's. Example: a(1)=24 because we have uuuuuv uuuuvu uuuvuu uuvuuu uvuuuu vuuuuu uuuuuz uuuuzu uuuzuu uuzuuu uzuuuu zuuuuu uuuuux uuuuxu uuuxuu uuxuuu uxuuuu xuuuuu uuuuuy uuuuyu uuuyuu uuyuuu uyuuuu yuuuuu. - Zerinvary Lajos, Jun 16 2008
Also convolution of A002457 with A020920, also convolution of A002697 with A038846, also convolution of A002802 with A020918, also convolution of A038845 with A038845. - Rui Duarte, Oct 08 2011

Links

Crossrefs

Cf. A038231.

Programs

  • GAP
    List([0..30], n-> 4^n*Binomial(n+5,5)); # G. C. Greubel, Jul 20 2019
  • Magma
    [4^n*Binomial(n+5, 5): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011
  • Maple
    seq(seq(binomial(i+5, j)*4^i, j =i), i=0..30); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+5,5)*4^n,n=0..30); # Zerinvary Lajos, Jun 16 2008
  • Mathematica
    CoefficientList[Series[1/(1-4x)^6,{x,0,30}],x] (* or *) LinearRecurrence[ {24,-240,1280,-3840,6144,-4096}, {1,24,336,3584,32256, 258048}, 30] (* Harvey P. Dale, Mar 24 2018 *)
  • PARI
    Vec(1/(1-4*x)^6 + O(x^30)) \\ Michel Marcus, Aug 21 2015
  • Sage
    [lucas_number2(n, 4, 0)*binomial(n,5)/2^10 for n in range(5, 35)] # Zerinvary Lajos, Mar 11 2009

Formula

a(n) = binomial(n+5, 5)*4^n.
G.f.: 1/(1-4*x)^6.
a(n) = Sum_{ i_1+i_2+i_3+i_4+i_5+i_6+i_7+i_8+i_9+i_10+i_11+i_12 = n} f(i_1)* f(i_2)*f(i_3)*f(i_4)*f(i_5)*f(i_6)*f(i_7)*f(i_8)*f(i_9)*f(i_10) *f(i_11)*f(i_12), with f(k)=A000984(k). - Rui Duarte, Oct 08 2011
E.g.f.: (15 + 120*x + 240*x^2 + 160*x^3 + 32*x^4)*exp(4*x)/3. - G. C. Greubel, Jul 20 2019
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 1620*log(4/3) - 465.
Sum_{n>=0} (-1)^n/a(n) = 12500*log(5/4) - 8365/3. (End)

A386271 Expansion of 1/(1 - 49*x)^(2/7).

Original entry on oeis.org

1, 14, 441, 16464, 662676, 27832392, 1201431588, 52862989872, 2359010923038, 106417603861492, 4842000975697886, 221851681068339504, 10223664969232645476, 473434331652157890504, 22014696421825341908436, 1027352499685182622393680, 48092938891512611510804145
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2025

Keywords

Crossrefs

Cf. A020918 (k=2, m=7), A020920 (k=2, m=9), A034835 (k=7, m=1), A034977 (k=8, m=1), A035024 (k=9, m=1), A216702 (k=4, m=3), A216703 (k=7, m=6), A354019 (k=6, m=1), this sequence (k=7, m=2), A386272 (k=7, m=3), A386273 (k=7, m=4), A386274 (k=7, m=5).

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/(1-49*x)^(2/7))

Formula

a(n) = (-49)^n * binomial(-2/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+2).
a(n) = 7^n * Product_{k=1..n} (7 - 5/k).
In general, 1/(1 - k^2*x)^(m/k) leads to the D-finite recurrence k*(k*n-k+m)*a(n-1) - n*a(n) = 0. This sequence is case k=7, m=2: (49*n-35)*a(n-1) - n*a(n) = 0. - Georg Fischer, Jul 19 2025

A002738 Coefficients for extrapolation.

Original entry on oeis.org

3, 60, 630, 5040, 34650, 216216, 1261260, 7001280, 37413090, 193993800, 981608628, 4867480800, 23728968900, 114011377200, 540972351000, 2538963567360, 11802213457650, 54396360988200, 248812984520100, 1130341536324000, 5103492036502860, 22913637714910800
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Let H be the n X n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row n-2 of B equals a(n-3). - T. D. Noe, May 01 2011

References

  • J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002802, A007744, A020918.
A diagonal of A331431.

Programs

  • Magma
    [3*Binomial(2*n+3,n)*Binomial(n+3,3): n in [0..30]]; // G. C. Greubel, Mar 21 2022
  • Mathematica
    Table[Total[Inverse[HilbertMatrix[n]][[n - 2]]], {n, 3, 25}] (* T. D. Noe, May 02 2011 *)
  • Sage
    [3*binomial(2*n+3,3)*binomial(2*n,n) for n in (0..30)] # G. C. Greubel, Mar 21 2022

Formula

From Alois P. Heinz, May 02 2011: (Start)
a(n) = 3*binomial(2*n+3,n)*binomial(n+3,n).
G.f.: 3*(1 + 6*x)/(1-4*x)^(7/2). (End)
a(n) = binomial(2*n+3,n)*(n^3 + 6*n^2 + 11*n+6)/2. - Charles R Greathouse IV, May 02 2011
a(n) = 3*A007744(n). - R. J. Mathar, Jan 21 2020
a(n) = (3/2)*( 5*A020918(n) - 3*A002802(n)). - G. C. Greubel, Mar 21 2022

Extensions

Extended by T. D. Noe, May 01 2011

A382332 Expansion of 1/(1 - 4*x/(1-x)^2)^(7/2).

Original entry on oeis.org

1, 14, 154, 1470, 12866, 106078, 837018, 6385262, 47420674, 344553902, 2458367898, 17272647966, 119770278978, 821068784382, 5572735854234, 37490757508302, 250247764120578, 1658681038111566, 10924592141535898, 71541334475749502, 466060971286552642
Offset: 0

Views

Author

Seiichi Manyama, Mar 30 2025

Keywords

Links

Crossrefs

Cf. A110170, A377198, A382274.
Cf. A020918, A377200.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - 4*x/(1-x)^2)^(7/2))); // Vincenzo Librandi, May 12 2025
  • Mathematica
    Table[Sum[(-4)^k* Binomial[-7/2,k]*Binomial[n+k-1, n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, May 12 2025 *)
  • PARI
    a(n) = sum(k=0, n, (-4)^k*binomial(-7/2, k)*binomial(n+k-1, n-k));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (7-5*k/n) * (n-k) * a(k).
a(n) = ((7*n+7)*a(n-1) - (7*n-28)*a(n-2) + (n-3)*a(n-3))/n for n > 2.
a(n) = Sum_{k=0..n} (-4)^k * binomial(-7/2,k) * binomial(n+k-1,n-k).
a(n) = 14*n*hypergeom([9/2, 1-n, 1+n], [3/2, 2], -1) for n > 0. - Stefano Spezia, Mar 30 2025
a(n) ~ 2^(5/4) * (1 + sqrt(2))^(2*n) * n^(5/2) / (15*sqrt(Pi)). - Vaclav Kotesovec, May 03 2025

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
Previous Showing 11-15 of 15 results.

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