A202824 -id:A202824 - cp's OEIS frontend

A294042 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp((1+x)^k - 1).

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 15, 20, 1, 0, 1, 5, 28, 87, 76, 1, 0, 1, 6, 45, 232, 585, 312, 1, 0, 1, 7, 66, 485, 2248, 4383, 1384, 1, 0, 1, 8, 91, 876, 6145, 24544, 35919, 6512, 1, 0, 1, 9, 120, 1435, 13716, 88245, 295456, 318195, 32400, 1, 0

Offset: 0

Seiichi Manyama, Oct 22 2017

			Square array A(n,k) begins:
   1, 1,   1,    1,     1,     1, ...
   0, 1,   2,    3,     4,     5, ...
   0, 1,   6,   15,    28,    45, ...
   0, 1,  20,   87,   232,   485, ...
   0, 1,  76,  585,  2248,  6145, ...
   0, 1, 312, 4383, 24544, 88245, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..5 give A000007, A000012, A000898, A192989, A202824, A202825.
Rows n=0..2 give A000012, A001477, A000384.
Main diagonal gives A294045.
Cf. A000110, A293991.

A(0,k) = 1 and A(n,k) = k * (n-1)! * Sum_{j=1..min(k,n)} binomial(k-1,j-1) * A(n-j,k)/(n-j)! for n > 0.

A(n,k) = Sum_{j=0..n} k^j * Stirling1(n,j) * Bell(j). - Seiichi Manyama, Jan 31 2024

A202825 Expansion of e.g.f.: exp( (1+x)^5 - 1 ).

Original entry on oeis.org

1, 5, 45, 485, 6145, 88245, 1403725, 24383525, 457473825, 9191615525, 196455592525, 4442277025125, 105787516038625, 2642880807687125, 69040011233566125, 1880443426122681125, 53268012941536530625, 1565875625728027213125, 47673392561258073158125

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 5*x + 45*x^2/2! + 485*x^3/3! + 6145*x^4/4! +...
where A(x) = exp(5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5).

Seiichi Manyama, Table of n, a(n) for n = 0..467

Cf. A000110, A000898, A008275, A192989, A202824.

List([0..20], n-> Sum([0..n], k-> (-1)^(n-k)*5^k*Bell(k)* Stirling1(n,k) )); # G. C. Greubel, Jul 25 2019

[(&+[5^k*Bell(k)*StirlingFirst(n,k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jul 25 2019

Table[Sum[StirlingS1[n, k] 5^k BellB[k], {k, 0, n}], {n, 0, 20}] (* Emanuele Munarini, Sep 06 2017 *)

makelist(sum(stirling1(n,k)*5^k*belln(k),k,0,n),n,0,12); /* Emanuele Munarini, Sep 06 2017 */

{a(n)=n!*polcoeff(exp((1+x +x*O(x^n))^5-1),n)}

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k) * 5^k)}

[sum((-1)^(n-k)*5^k*bell_number(k)*stirling_number1(n,k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Jul 25 2019

a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(k) * 5^k.

a(n+5) - 5*a(n+4) - 20*(n+4)*a(n+3) - 30*(n+3)*(n+4)*a(n+2) - 20*(n+2)*(n+3)*(n+4)* a(n+1) - 5*(n+1)*(n+2)*(n+3)*(n+4)*a(n) = 0. - Emanuele Munarini, Sep 06 2017

A202826 E.g.f.: exp( 1/(1-x)^3 - 1 ).

Original entry on oeis.org

1, 3, 21, 195, 2241, 30483, 476469, 8383203, 163532385, 3496040163, 81159271029, 2030708891907, 54427341596769, 1554460972941555, 47097454520401749, 1507969940021725347, 50850987639474121281, 1800630391669594010307, 66775808799868618561365

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 3*x + 21*x^2/2! + 195*x^3/3! + 2241*x^4/4! +...
where
log(A(x)) = 3*x + 6*x^2 + 10*x^3 + 15*x^4 + 21*x^5 + 28*x^6 +...

Seiichi Manyama, Table of n, a(n) for n = 0..420 (terms 0..200 from Vincenzo Librandi)

Cf. A000262, A136658, A202824.

CoefficientList[Series[E^(1/(1-x)^3-1), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 12 2013 *)
Table[Sum[Abs[StirlingS1[n, k]] 3^k BellB[k], {k, 0, n}], {n, 0, 100}] (* Emanuele Munarini, Sep 01 2017 *)

makelist(sum(abs(stirling1(n,k))*3^k*belln(k),k,0,n),n,0,12); /* Emanuele Munarini, Sep 01 2017 */

{a(n)=n!*polcoeff(exp(1/(1-x +x*O(x^n))^3-1),n)}

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k) *(-1)^(n-k)*3^k)}

a(n) = Sum_{k=0..n} Stirling1(n, k) * Bell(k) * (-1)^(n-k) * 3^k.
a(n) ~ n! * 1/2*3^(1/8)*exp(sqrt(3*n)/2 -3/4 + (3*n)^(1/4)*(4/3*sqrt(n) + 5/24*sqrt(3)) )/(sqrt(2*Pi)*n^(5/8)) * (1 + 871/2560*(3/n)^(1/4)). - Vaclav Kotesovec, Feb 12 2013
a(n+4) - (4*n+15)*a(n+3) + 6*(n+2)*(n+3)*a(n+2) - 4*(n+1)*(n+2)+(n+3)*a(n+1) + n*(n+1)*(n+2)*(n+3)*a(n) = 0. - Emanuele Munarini, Sep 01 2017

Example corrected by Vaclav Kotesovec, Feb 12 2013

A369752 Expansion of e.g.f. exp(1 - (1+x)^4).

Original entry on oeis.org

1, -4, 4, 56, -104, -2464, 1696, 181184, 462016, -17069824, -141580544, 1593913856, 33015560704, -47193585664, -6973651011584, -50207289585664, 1214484253413376, 25500259291480064, -72069247145590784, -8696105637665603584, -81680899029758541824

Offset: 0

Seiichi Manyama, Jan 30 2024

sign

Column k=4 of A369738.

Cf. A000587, A202824.

With[{nn=20},CoefficientList[Series[Exp[1-(1+x)^4],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 29 2024 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(1-(1+x)^4)))

a(0) = 1; a(n) = -4 * (n-1)! * Sum_{k=1..min(4,n)} binomial(3,k-1) * a(n-k)/(n-k)!.
a(n) = Sum_{k=0..n} 4^k * Stirling1(n,k) * A000587(k).
D-finite with recurrence a(n) +4*a(n-1) +12*(n-1)*a(n-2) +12*(n-1)*(n-2)*a(n-3) +4*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Feb 02 2024

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A294042 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp((1+x)^k - 1).

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A202825 Expansion of e.g.f.: exp( (1+x)^5 - 1 ).

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A202826 E.g.f.: exp( 1/(1-x)^3 - 1 ).

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A369752 Expansion of e.g.f. exp(1 - (1+x)^4).

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