A002544 -id:A002544 - cp's OEIS frontend

A111941 Matrix log of triangle A111940, which shifts columns left and up under matrix inverse; these terms are the result of multiplying each element in row n and column k by (n-k)!.

0, 1, 0, -1, -1, 0, 1, 1, 1, 0, -2, -1, -1, -1, 0, 4, 2, 1, 1, 1, 0, -12, -4, -2, -1, -1, -1, 0, 36, 12, 4, 2, 1, 1, 1, 0, -144, -36, -12, -4, -2, -1, -1, -1, 0, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0, 14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0, -86400, -14400, -2880, -576, -144, -36

Offset: 0

Paul D. Hanna, Aug 23 2005

			Triangle begins:
0;
1, 0;
-1, -1, 0;
1, 1, 1, 0;
-2, -1, -1, -1, 0;
4, 2, 1, 1, 1, 0;
-12, -4, -2, -1, -1, -1, 0;
36, 12, 4, 2, 1, 1, 1, 0;
-144, -36, -12, -4, -2, -1, -1, -1, 0;
576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0;
14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-86400, -14400, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0;
518400, 86400, 14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
-3628800, -518400, -86400, -14400, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0; ...
where, apart from signs, the columns are all the same (A111942).
...
Triangle A111940 begins:
1;
1, 1;
-1, -1, 1;
0, 0, 1, 1;
0, 0, -1, -1, 1;
0, 0, 0, 0, 1, 1;
0, 0, 0, 0, -1, -1, 1;
0, 0, 0, 0, 0, 0, 1 ,1;
0, 0, 0, 0, 0, 0, -1, -1, 1; ...
where the matrix inverse shifts columns left and up one place.
...
The matrix log of A111940, with factorial denominators, begins:
0;
1/1!, 0;
-1/2!, -1/1!, 0;
1/3!, 1/2!, 1/1!, 0;
-2/4!, -1/3!, -1/2!, -1/1!, 0;
4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-144/8!, -36/7!, -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
576/9!, 144/8!, 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
-2880/10!, -576/9!, -144/8!, -36/7!, -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
14400/11!, 2880/10!, 576/9!, 144/8!, 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0; ...
Note that the square of the matrix log of A111940 begins:
0;
0, 0;
-1, 0, 0;
0, -1, 0, 0;
-1/12, 0, -1, 0, 0;
0, -1/12, 0, -1, 0, 0;
-1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
0, -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
-1/16632, 0, -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0; ...
where nonzero terms are negative unit fractions with denominators given by A002544:
[1, 12, 90, 560, 3150, 16632, 84084, 411840, ...,  C(2*n+1,n)*(n+1)^2, ...].

Cf. A111940 (triangle), A111942 (column 0), A110504 (variant).

{T(n,k,q=-1) = local(A=Mat(1),B); if(n

T(n, k) = (-1)^k*T(n-k, 0) = (-1)^k*A111942(n-k) for n>=k>=0.

A175478 Decimal expansion of log(3)^2.

Original entry on oeis.org

1, 2, 0, 6, 9, 4, 8, 9, 6, 0, 8, 1, 2, 5, 8, 1, 9, 7, 7, 8, 4, 3, 7, 7, 9, 1, 2, 3, 8, 4, 9, 3, 6, 5, 9, 1, 3, 6, 1, 8, 4, 6, 3, 3, 4, 6, 6, 2, 9, 2, 2, 1, 9, 8, 4, 8, 1, 6, 7, 2, 6, 8, 4, 0, 0, 5, 8, 2, 1, 5, 5, 1, 4, 8, 0, 7, 9, 8, 5, 2, 5, 4, 4, 5, 8, 5, 4, 4, 3, 0, 1, 7, 7, 1, 4, 0, 9, 3, 3, 3, 4, 2, 2, 8, 3

Offset: 1

R. J. Mathar, May 25 2010

			1.2069489608125819778437...

Index entries for transcendental numbers

Cf. A002391, A253191.

RealDigits[Log[3]^2,10,120][[1]] (* Harvey P. Dale, Jan 23 2012 *)

log(3)^2 \\ Charles R Greathouse IV, May 15 2019

Equals A002391^2.

Equals Sum_{n >= 0} (-1)^n*(4/3)^(n+1)/((n+1)^2*binomial(2*n+1,n)). See my entry in A002544 dated Apr 18 2017. - Peter Bala, Jan 30 2023

A245111 G.f.: A(x,y) = Sum_{n>=0} exp(-y/(1-nx)) y^n/(1-n*x)^n / n!.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 12, 10, 0, 1, 35, 90, 35, 0, 1, 90, 525, 560, 126, 0, 1, 217, 2520, 5460, 3150, 462, 0, 1, 504, 10836, 42000, 46200, 16632, 1716, 0, 1, 1143, 43470, 280665, 519750, 342342, 84084, 6435, 0, 1, 2550, 166375, 1709400, 4969965, 5297292, 2312310, 411840, 24310

Offset: 0

Paul D. Hanna, Jul 12 2014

Compare g.f. to: 1/(1-x*y) = Sum_{n>=0} exp(-y*(1+n*x)) * y^n*(1+n*x)^n / n!.
Row sums equal A245110.
Antidiagonal sums: A218667.
Main diagonal is: C(2*n-1,n) (A001700).
Secondary diagonal: C(2*n-1,n)*n^2 (A002544).

			G.f.: A(x,y) = 1 + x*y + x^2*(y + 3*y^2)
+ x^3*(y + 12*y^2 + 10*y^3)
+ x^4*(y + 35*y^2 + 90*y^3 + 35*y^4)
+ x^5*(y + 90*y^2 + 525*y^3 + 560*y^4 + 126*y^5)
+ x^6*(y + 217*y^2 + 2520*y^3 + 5460*y^4 + 3150*y^5 + 462*y^6) +...
where
A(x,y) = exp(-y) + exp(-y/(1-x))*y/(1-x) + (exp(-y/(1-2*x))*y^2/(1-2*x)^2)/2!
+ (exp(-y/(1-3*x))*y^3/(1-3*x)^3)/3! + (exp(-y/(1-4*x))*y^4/(1-4*x)^4)/4!
+ (exp(-y/(1-5*x))*y^5/(1-5*x)^5)/5! + (exp(-y/(1-6*x))*y^6/(1-6*x)^6)/6!
+ (exp(-y/(1-7*x))*y^7/(1-7*x)^7)/7! + (exp(-y/(1-8*x))*y^8/(1-8*x)^8)/8! +...
simplifies to a power series with only integer coefficients of x^n*y^k.
Triangle begins:
1;
0, 1;
0, 1, 3;
0, 1, 12, 10;
0, 1, 35, 90, 35;
0, 1, 90, 525, 560, 126;
0, 1, 217, 2520, 5460, 3150, 462;
0, 1, 504, 10836, 42000, 46200, 16632, 1716;
0, 1, 1143, 43470, 280665, 519750, 342342, 84084, 6435;
0, 1, 2550, 166375, 1709400, 4969965, 5297292, 2312310, 411840, 24310;
0, 1, 5621, 615780, 9754030, 42567525, 68549481, 47087040, 14586000, 1969110, 92378; ...
where T(n,k) = A048993(n,k) * C(n+k-1, k-1) for k>0.

Paul D. Hanna, Table of n, a(n), of flattened triangle for rows 0..32

Cf. A245110, A218667, A001700, A002544, A048993.

/* From definition: */
{T(n,k)=local(A=1+x*y); A=sum(k=0, n, 1/(1-k*x+x*O(x^n))^k*y^k/k!*exp(-y/(1-k*x+x*O(x^n))+y*O(y^n))); polcoeff(polcoeff(A, n,x),k,y)}
for(n=0, 10, for(k=0,n, print1(T(n,k),", "));print(""))

/* From T(n,k) = Stirling2(n, k) * C(n+k-1, k-1) */
{Stirling2(n, k) = sum(j=0, k, (-1)^(k+j) * binomial(k, j) * j^n) / k!}
{T(n,k)=if(k==0,0^n,Stirling2(n, k) * binomial(n+k-1, k-1))}
for(n=0, 10, for(k=0,n, print1(T(n,k),", "));print(""))

T(n,k) = Stirling2(n, k) * binomial(n+k-1, k-1) for k>0, where Stirling2(n,k) = A048993(n,k).

A106440 a(n) = binomial(2n+4,n)*binomial(n+4,4).

Original entry on oeis.org

1, 30, 420, 4200, 34650, 252252, 1681680, 10501920, 62355150, 355655300, 1963217256, 10546208400, 55367594100, 285028443000, 1442592936000, 7193730107520, 35406640372950, 172255143129300, 829376615067000

Offset: 0

Paul Barry, May 02 2005

Fifth column of A104684.

Diagonal of the rational function 1 / (1 - x - y)^5. - Ilya Gutkovskiy, Apr 24 2025

Michael De Vlieger, Table of n, a(n) for n = 0..1642
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.

Cf. A007744, A002544, A002457.

Cf. A002803, A020920.

Table[Binomial[2n+4,n]Binomial[n+4,4],{n,0,20}] (* Harvey P. Dale, May 03 2019 *)

G.f.: (1+12x+6x^2)/(1-4x)^(9/2).
D-finite with recurrence n^2*a(n) -2*(n+2)*(2*n+3)*a(n-1)=0. - R. J. Mathar, Feb 20 2015
G.f.: 2F1(5/2,3;1;4x). - R. J. Mathar, Aug 09 2015
a(n) = A020920(n)+12*A020920(n-1)+6*A020920(n-2). - R. J. Mathar, Aug 09 2015
a(n) = (n+1)*A002803(n). - R. J. Mathar, Aug 09 2015

A349427 a(n) = ((n+1)^2 - 2) * binomial(2*n-2,n-1) / 2.

Original entry on oeis.org

0, 1, 7, 42, 230, 1190, 5922, 28644, 135564, 630630, 2892890, 13117676, 58903572, 262303132, 1159666900, 5094808200, 22259364120, 96773942790, 418882316490, 1805951924700, 7758285404100, 33221013445620, 141830949914940, 603876402587640, 2564713671647400

Offset: 0

Ilya Gutkovskiy, Nov 17 2021

Cf. A000108, A000217, A000984, A001700, A001793, A002457, A002544, A008865, A037966, A088218, A127736.

Table[((n + 1)^2 - 2) Binomial[2 n - 2, n - 1]/2, {n, 0, 24}]
nmax = 24; CoefficientList[Series[x (1 - x) (1 - 2 x)/(1 - 4 x)^(5/2), {x, 0, nmax}], x]

a(n) = ((n+1)^2 - 2) * binomial(2*n-2,n-1) / 2 \\ Andrew Howroyd, Nov 20 2021

G.f.: x * (1 - x) * (1 - 2*x) / (1 - 4*x)^(5/2).
E.g.f.: x * exp(2*x) * (2 * (1 + x) * BesselI(0,2*x) + (1 + 2*x) * BesselI(1,2*x)) / 2.
a(n) = n * ((n+1)^2 - 2) * Catalan(n-1) / 2.
a(n) = Sum_{k=0..n} binomial(n,k)^2 * A000217(k).
a(n) ~ 2^(2*n-3) * n^(3/2) / sqrt(Pi).
D-finite with recurrence (n-1)*(n^2-2)*a(n) -2*(2*n-3)*(n^2+2*n-1)*a(n-1)=0. - R. J. Mathar, Mar 06 2022

A228080 (5n+2)!/(2(n!)^5), n >= 0.

Original entry on oeis.org

1, 2520, 7484400, 22870848000, 70579794285000, 218799620836917120, 679953587124305894400, 2116187746296592370688000, 6592431144164903462359935000, 20550499897066845200729434200000, 64091912654977017603465324370118400, 199956261330234671205699024876891648000

Offset: 0

Karol A. Penson, Aug 09 2013

nonn

Although limit( a(n)^(1/n), n=infinity ) = 5^5, apparently this sequence is not a Hausdorff moment sequence of any positive function on (0,5^5).

Cf. A002544.

Maple
```
seq((5*n+2)!/(2*(n!)^5), n=0..11).
```

Table[(5n+2)!/(2(n!)^5),{n,0,15}] (* Harvey P. Dale, Aug 04 2019 *)

In Maple notation:
O.g.f. : hypergeom([3/5, 4/5, 6/5, 7/5], [1, 1, 1], 5^5*z);
E.g.f. : hypergeom([3/5, 4/5, 6/5, 7/5], [1, 1, 1, 1], 5^5*z);
Asymptotics: a(n) -> (25*n^2+5*n-2)*(5^(5*n+1/2))* n^(-2)/(8*Pi^2), for n -> infinity.
D-finite with recurrence (n^4)*a(n) -5*(5*n+1)*(5*n+2)*(5*n-2)*(5*n-1)*a(n-1)=0. - R. J. Mathar, Jul 27 2022

cp's OEIS Frontend

A111941 Matrix log of triangle A111940, which shifts columns left and up under matrix inverse; these terms are the result of multiplying each element in row n and column k by (n-k)!.

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A175478 Decimal expansion of log(3)^2.

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A245111 G.f.: A(x,y) = Sum_{n>=0} exp(-y/(1-n*x)) * y^n/(1-n*x)^n / n!.

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A106440 a(n) = binomial(2n+4,n)*binomial(n+4,4).

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A349427 a(n) = ((n+1)^2 - 2) * binomial(2*n-2,n-1) / 2.

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A228080 (5*n+2)!/(2*(n!)^5), n >= 0.

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Maple

Mathematica

Formula

A245111 G.f.: A(x,y) = Sum_{n>=0} exp(-y/(1-nx)) y^n/(1-n*x)^n / n!.

A228080 (5n+2)!/(2(n!)^5), n >= 0.