A202832 -id:A202832 - cp's OEIS frontend

A202831 Expansion of e.g.f.: exp(4x/(1-5x)) / sqrt(1-25*x^2).

1, 4, 81, 1444, 44521, 1397124, 58354321, 2574344644, 136043683281, 7657406908804, 489836445798001, 33351743794661604, 2504378700538997881, 199445618093659242244, 17189578072429077875121, 1564487078400498014277124, 152146464623361858013314721

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: 1 + 4*x + 81*x^2/2! + 1444*x^3/3! + 44521*x^4/4! + 1397124*x^5/5! + ...
where A(x) = 1 + 2^2*x + 9^2*x^2/2! + 38^2*x^3/3! + 211^2*x^4/4! + 1182^2*x^5/5! + ... + A202832(n)^2*x^n/n! + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A202832, A202827, A202828, A202829, A202833, A202835, A202836.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(4*x/(1-5*x))/Sqrt(1-25*x^2) ))); // G. C. Greubel, Jun 21 2022

CoefficientList[Series[Exp[4*x/(1-5*x)]/Sqrt[1-25*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

{a(n)=n!*polcoeff(exp(4*x/(1-5*x)+x*O(x^n))/sqrt(1-25*x^2+x*O(x^n)),n)}

{a(n)=n!^2*polcoeff(exp(2*x+5*x^2/2+x*O(x^n)),n)^2}

{a(n)=sum(k=0,n\2,2^(n-3*k)*5^k*n!/((n-2*k)!*k!))^2}

def A202831_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-5*x))/sqrt(1-25*x^2) ).egf_to_ogf().list()
A202831_list(40) # G. C. Greubel, Jun 21 2022

a(n) = A202832(n)^2, where the e.g.f. of A202832 is exp(2*x + 5*x^2/2).
a(n) = ( Sum_{k=0..[n/2]} 2^(n-3*k)*5^k * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(4*sqrt(n/5)-2/5-n)*5^n/2. - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (5*n-1)*a(n-1) + 5*(n-1)*(5*n-1)*a(n-2) - 125*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A376826 Array read by antidiagonals: T(n,k) = n! * [x^n] exp(2x + (k/2)x^2), n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 5, 8, 1, 2, 6, 14, 16, 1, 2, 7, 20, 43, 32, 1, 2, 8, 26, 76, 142, 64, 1, 2, 9, 32, 115, 312, 499, 128, 1, 2, 10, 38, 160, 542, 1384, 1850, 256, 1, 2, 11, 44, 211, 832, 2809, 6512, 7193, 512, 1, 2, 12, 50, 268, 1182, 4864, 15374, 32400, 29186, 1024

Offset: 0

Andrew Howroyd, Oct 07 2024

			Array begins:
======================================================
n\k |   0    1    2     3     4     5     6      7 ...
----+-------------------------------------------------
  0 |   1    1    1     1     1     1     1      1 ...
  1 |   2    2    2     2     2     2     2      2 ...
  2 |   4    5    6     7     8     9    10     11 ...
  3 |   8   14   20    26    32    38    44     50 ...
  4 |  16   43   76   115   160   211   268    331 ...
  5 |  32  142  312   542   832  1182  1592   2062 ...
  6 |  64  499 1384  2809  4864  7639 11224  15709 ...
  7 | 128 1850 6512 15374 29696 50738 79760 118022 ...
     ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Arvind Ayyer, Hiranya Kishore Dey and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv preprint arXiv:2406.06036, [math.RT], 2024.

Columns 0..5 are A000079, A005425, A000898, A202830, A193778, A202832.

Cf. A359762, A373625.

T(n,k) = {sum(i=0, n\2, binomial(n,2*i) * 2^(n-2*i) * k^i * (2*i)!/(2^i*i!))}

E.g.f. of column k: exp(2*x + k*x^2/2).
Column k is the binomial transform of column k of A359762.
T(n,k) = Sum_{i=0..floor(n/2)} binomial(n,2*i) * 2^(n-2*i) * k^i * (2*i-1)!!.
T(n,k) = Sum_{i=0..floor(n/2)} 2^(n-3*i) * k^i * n! / ((n-2*i)! * i!).

cp's OEIS Frontend

A202831 Expansion of e.g.f.: exp(4x/(1-5x)) / sqrt(1-25*x^2).

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A376826 Array read by antidiagonals: T(n,k) = n! * [x^n] exp(2x + (k/2)x^2), n >= 0, k >= 0.

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cp's OEIS Frontend

A202831 Expansion of e.g.f.: exp(4*x/(1-5*x)) / sqrt(1-25*x^2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

SageMath

Formula

A376826 Array read by antidiagonals: T(n,k) = n! * [x^n] exp(2*x + (k/2)*x^2), n >= 0, k >= 0.

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Formula

A202831 Expansion of e.g.f.: exp(4x/(1-5x)) / sqrt(1-25*x^2).

A376826 Array read by antidiagonals: T(n,k) = n! * [x^n] exp(2x + (k/2)x^2), n >= 0, k >= 0.