A220181 -id:A220181 - cp's OEIS frontend

A243809 G.f.: exp( Integral Sum_{n>=1} n!n^nx^(n-1) / Product_{k=1..n} (1+knx) dx ).

1, 1, 4, 42, 909, 33969, 1948514, 158770640, 17419561466, 2474812055850, 441910422152592, 96867456432497772, 25572186966552515130, 8002470602289313981938, 2929213328377453597523820, 1239933908245021946285387592, 601020049737946926794959094457

Offset: 0

Paul D. Hanna, Jun 11 2014

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 42*x^3 + 909*x^4 + 33969*x^5 + 1948514*x^6 +...
where the logarithmic derivative is given by the series:
A'(x)/A(x) = 1/(1+x) + 2!*2^2*x/((1+1*2*x)*(1+2*2*x)) + 3!*3^3*x^2/((1+1*3*x)*(1+2*3*x)*(1+3*3*x)) + 4!*4^4*x^3/((1+1*4*x)*(1+2*4*x)*(1+3*4*x)*(1+4*4*x)) + 5!*5^5*x^4/((1+1*5*x)*(1+2*5*x)*(1+3*5*x)*(1+4*5*x)*(1+5*5*x)) +...
Explicitly,
A'(x)/A(x) = 1 + 7*x + 115*x^2 + 3451*x^3 + 164731*x^4 + 11467387*x^5 +...+ A220181(n+1)*x^n +...
compare to:
G(x) = x + 7*x^2/2! + 115*x^3/3! + 3451*x^4/4! + 164731*x^5/5! + 11467387*x^6/6! +...+ A220181(n)*x^n/n! +...
where G(x) = (1-exp(-x)) + (1-exp(-2*x))^2 + (1-exp(-3*x))^3 + (1-exp(-4*x))^4 +...

Cf. A220181, A243807.

{a(n)=local(A=1+x); A=exp(intformal(sum(m=1, n+1, m^m*m!*x^(m-1)/prod(k=1, m, 1+m*k*x +x*O(x^n))))); polcoeff(A,n)}
for(n=0, 20, print1(a(n), ", "))

/* From g.f. exp( Sum_{n>=1} A220181(n)*x^n/n ): */
{A220181(n)=n!*polcoeff(sum(k=0, n, (1-exp(-k*x+x*O(x^n)))^k), n)}
{a(n)=polcoeff(exp(sum(m=1,n,A220181(m)*x^m/m) +x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} A220181(n)*x^n/n ) where Sum_{n>=1} A220181(n)*x^n/n! = Sum_{n>=1} (1 - exp(-n*x))^n.

A201007 G.f.: 1/(1-x) = Sum_{n>=0} a(n)x^n / Product_{k=1..n} (1 + nk*x).

Original entry on oeis.org

1, 1, 2, 12, 162, 4020, 161190, 9580200, 794045490, 87732586200, 12482467492950, 2225389826721600, 486286707998356650, 127896148968309802080, 39873063650831725704390, 14545617596016448962820800, 6140116931023810866657175650, 2970359726329509983655533867520

Offset: 0

Paul D. Hanna, Jan 08 2013

nonn

Compare to a g.f. involving the factorial numbers:
1/(1-x) = Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 + k*x).
Also compare to a g.f. of A220181:
Sum_{n>=0} n^n * n! * x^n / Product_{k=1..n} (1 + n*k*x).

			1/(1-x) = 1 + x/(1+x) + 2*x^2/((1+2*1*x)*(1+2*2*x)) + 12*x^3/((1+3*1*x)*(1+3*2*x)*(1+3*3*x)) + 162*x^4/((1+4*1*x)*(1+4*2*x)*(1+4*3*x)*(1+4*4*x)) + 4020*x^5/((1+5*1*x)*(1+5*2*x)*(1+5*3*x)*(1+5*4*x)*(1+5*5*x)) +...

{a(n)=if(n==0, 1, 1-polcoeff(sum(k=0, n-1, a(k)*x^k/prod(j=1, k, 1+k*j*x+x*O(x^n))), n))}
for(n=0, 20, print1(a(n), ", "))

A347940 Array T(n, k) = Sum_{j=2..n+2} (-1)^(n-j)Stirling2(n+1, j-1)j!*j^k/2, for n and k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 4, 7, 4, 8, 23, 23, 8, 16, 73, 115, 73, 16, 32, 227, 533, 533, 227, 32, 64, 697, 2359, 3451, 2359, 697, 64, 128, 2123, 10133, 20753, 20753, 10133, 2123, 128, 256, 6433, 42655, 118843, 164731, 118843, 42655, 6433, 256, 512, 19427, 177053, 657833, 1220657, 1220657, 657833, 177053, 19427, 512

Offset: 0

Michel Marcus, Sep 20 2021

T(m, n) is the number of saturated Cp^m*q^n-transfer systems where Cp^m*q^n is the cyclic group of order p^m*q^n, for m, n >= 0, p and q primes. See Hafeez et al. link page 1.

			Array begins:
   1   2     4      8      16       32 ...
   2   7    23     73     227      697 ...
   4  23   115    533    2359    10133 ...
   8  73   533   3451   20753   118843 ...
  16 227  2359  20753  164731  1220657 ...
  32 697 10133 118843 1220657 11467387 ...
  ...

Arvind Ayyer and Beáta Bényi, Toppling on permutations with an extra chip, El. J. Comb., 28 (2021), P4.18. The array seems to appear in Table 6.
Usman Hafeez, Peter Marcus, Kyle Ormsby and Angélica Osorno, Saturated and linear isometric transfer systems for cyclic groups of order p^m*q^n, arXiv:2109.08210 [math.AT], 2021.

Columns k=0-1 gives A000079, A083313(n+1).
Main diagonal gives A220181(n+1).
Cf. A008277 (Stirling2), A143494.

T(n, k) = sum(j=2, n+2, (-1)^(n-j)*stirling(n+1, j-1, 2)*j!*j^k/2);

T(n,k) = T(k,n).

T(n,k) = Sum_{j=0..min(n,k)} (j!*(j+2)!/2)*Stirling2(n+2,j+2;2)*Stirling2(k+2,j+2;2), n,k >= 0, where Stirling2(n,k;2) are the 2-Stirling numbers of the second kind A143494. - Fabián Pereyra, Jan 08 2022

cp's OEIS Frontend

A243809 G.f.: exp( Integral Sum_{n>=1} n!n^nx^(n-1) / Product_{k=1..n} (1+knx) dx ).

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A201007 G.f.: 1/(1-x) = Sum_{n>=0} a(n)x^n / Product_{k=1..n} (1 + nk*x).

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PARI

A347940 Array T(n, k) = Sum_{j=2..n+2} (-1)^(n-j)Stirling2(n+1, j-1)j!*j^k/2, for n and k >= 0, read by antidiagonals.

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PARI

Formula

cp's OEIS Frontend

A243809 G.f.: exp( Integral Sum_{n>=1} n!*n^n*x^(n-1) / Product_{k=1..n} (1+k*n*x) dx ).

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Author

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Examples

Crossrefs

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PARI

PARI

Formula

A201007 G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n / Product_{k=1..n} (1 + n*k*x).

Views

Author

Keywords

Comments

Examples

Programs

PARI

A347940 Array T(n, k) = Sum_{j=2..n+2} (-1)^(n-j)*Stirling2(n+1, j-1)*j!*j^k/2, for n and k >= 0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A243809 G.f.: exp( Integral Sum_{n>=1} n!n^nx^(n-1) / Product_{k=1..n} (1+knx) dx ).

A201007 G.f.: 1/(1-x) = Sum_{n>=0} a(n)x^n / Product_{k=1..n} (1 + nk*x).

A347940 Array T(n, k) = Sum_{j=2..n+2} (-1)^(n-j)Stirling2(n+1, j-1)j!*j^k/2, for n and k >= 0, read by antidiagonals.