A192989 -id:A192989 - 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

A202824 Expansion of e.g.f.: exp( (1+x)^4 - 1 ).

Original entry on oeis.org

1, 4, 28, 232, 2248, 24544, 295456, 3869632, 54555328, 821239552, 13115934976, 221076780544, 3915685846528, 72609585620992, 1405168845395968, 28302270409560064, 591874919018500096, 12824294700196052992, 287350628454224478208, 6647086535396002004992

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 4*x + 28*x^2/2! + 232*x^3/3! + 2248*x^4/4! +...
where A(x) = exp(4*x + 6*x^2 + 4*x^3 + x^4).

Seiichi Manyama, Table of n, a(n) for n = 0..501

Cf. A000110, A000898, A008275, A192989, A202825.

List([0..20], n-> Sum([0..n], k-> (-1)^(n-k)*4^k*Bell(k)* Stirling1(n,k) )); # G. C. Greubel, Jul 25 2019

[(&+[4^k*Bell(k)*StirlingFirst(n,k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jul 25 2019

CoefficientList[Series[Exp[(1+x)^4-1], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)
Table[Sum[ (-1)^(n - k) Abs[StirlingS1[n, k]] 4^k BellB[k], {k, 0, n}], {n, 0, 100}] (* Emanuele Munarini, Aug 31 2017 *)

a(n) := sum((-1)^(n-k)*abs(stirling1(n,k))*4^k*belln(k),k,0,n);
makelist(a(n),n,0,12); /* Emanuele Munarini, Aug 31 2017 */

{a(n)=n!*polcoeff(exp((1+x +x*O(x^n))^4-1),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) * 4^k)}

[sum((-1)^(n-k)*4^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) * 4^k.
a(n) ~ n^(3*n/4)*2^(n/2-1)*exp(-3/4+5/16*sqrt(2)*n^(1/4)+sqrt(2)*n^(3/4)-3/4*n+3/4*sqrt(n)). - Vaclav Kotesovec, May 23 2013
a(n+4) - 4*a(n+3) - 12*(n+3)*a(n+2) - 12*(n+2)*(n+3)*a(n+1) - 4*(n+1)*(n+2)*(n+3)*a(n) = 0. - Emanuele Munarini, Aug 31 2017

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

A369751 Expansion of e.g.f. exp(1 - (1+x)^3).

Original entry on oeis.org

1, -3, 3, 21, -63, -423, 1899, 15201, -72063, -832491, 3105459, 60090093, -110508543, -5224722831, -3828328677, 510699368313, 2104026859521, -52582823289171, -473592954347037, 5168227121231301, 92434892126557761, -357595962971807223, -17085974691782295477

Offset: 0

Seiichi Manyama, Jan 30 2024

sign

Column k=3 of A369738.

Cf. A000587, A192989.

A369751 := proc(n)
    option remember ;
    if n =0 then
        1;
    else
        add( binomial(2,k-1) * procname(n-k)/(n-k)!,k=1..min(3,n)) ;
        -3*(n-1)!*% ;
    end if;
end proc:
seq(A369751(n),n=0..20) ; # R. J. Mathar, Feb 02 2024

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(1-(1+x)^3)))

a(0) = 1; a(n) = -3 * (n-1)! * Sum_{k=1..min(3,n)} binomial(2,k-1) * a(n-k)/(n-k)!.
a(n) = Sum_{k=0..n} 3^k * Stirling1(n,k) * A000587(k).
D-finite with recurrence a(n) +3*a(n-1) +6*(n-1)*a(n-2) +3*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Feb 02 2024

A192667 E.g.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^3 - 1)^n/n!.

Original entry on oeis.org

1, 1, 6, 99, 2616, 95625, 4468608, 254426571, 17087348736, 1322490908817, 115902895680000, 11345706419279859, 1226971723559141376, 145275861381024623769, 18691551435638516649984, 2596726179631913433046875, 387404350400960574932385792

Offset: 0

Paul D. Hanna, Jul 13 2011

nonn

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 99*x^3/3! + 2616*x^4/4! +...
where (A(x) - 1)/exp(A(x)^3-1) = x
and A(x/G(x)) = 1 + x where G(x) = exp(3*x + 3*x^2 + x^3):
G(x) = 1 + 3*x + 15*x^2/2! + 87*x^3/3! + 585*x^4/4! + 4383*x^5/5! +...
Related expansions.
(A(x)^3-1) = 3*x + 24*x^2/2! + 411*x^3/3! + 11088*x^4/4!  + 410175*x^5/5! +...
(A(x)^3-1)^2 = 18*x^2/2! + 432*x^3/3! + 13320*x^4/4! + 529920*x^5/5! +...
(A(x)^3-1)^3 = 162*x^3/3! + 7776*x^4/4! + 377460*x^5/5! +...
(A(x)^3-1)^4 = 1944*x^4/4! + 155520*x^5/5! +...

Cf. A192949, A192989.

CoefficientList[1 + InverseSeries[Series[x/E^((1+x)^3 - 1), {x, 0, 20}], x],x]*Range[0, 20]! (* Vaclav Kotesovec, Feb 26 2014 *)

{a(n)=local(A=1+serreverse(x/exp(3*x+3*x^2+x^3+x^2*O(x^n))));n!*polcoeff(A,n)}

{a(n)=local(A=1+x); for(i=1, n, A=1+x*exp(A^3-1+x*O(x^n))); n!*polcoeff(A, n)}

{a(n)=local(A=1+x);for(i=1,n,A=1+x*sum(m=0,n,(A^3-1+x*O(x^n))^m/m!));n!*polcoeff(A,n)}

E.g.f. A(x) equals the formal inverse of function (x-1)/exp(x^3-1).
E.g.f. satisfies: A(x) = 1 + x*exp(A(x)^3-1).
E.g.f.: A(x) = 1 + Series_Reversion( x/exp((1+x)^3 - 1) ).
E.g.f. satisfies: A(x/G(x)) = 1 + x where G(x) = exp((1+x)^3 - 1) and G(x) = x/Series_Reversion(A(x)-1) = e.g.f. of A192989.
a(n) ~ n^(n-1) / (3*sqrt(s*(1+s^2)) * exp(n) * r^n), where s = 1/9*(3 + (297/2 - (81*sqrt(13))/2)^(1/3) + 3*((1/2)*(11 + 3*sqrt(13)))^(1/3)) = 1.2228950301592487561... is the root of the equation 3*(s-1)*s^2 = 1, and r = (s-1)*exp(1-s^3) = 0.097309376917122928890... - Vaclav Kotesovec, Feb 26 2014

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).

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A202824 Expansion of e.g.f.: exp( (1+x)^4 - 1 ).

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Maxima

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

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GAP

Magma

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A369751 Expansion of e.g.f. exp(1 - (1+x)^3).

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Maple

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A192667 E.g.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^3 - 1)^n/n!.

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Mathematica

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