A218667
O.g.f.: Sum_{n>=0} 1/(1-n*x)^n * x^n/n! * exp(-x/(1-n*x)).
Original entry on oeis.org
1, 0, 1, 1, 4, 13, 46, 181, 778, 3585, 17566, 91171, 499324, 2873839, 17317743, 108933098, 713481122, 4855161425, 34257461754, 250177938679, 1887886966690, 14699340919293, 117933068390123, 973776266303732, 8265721830953558, 72052688932613079, 644393453082317301
Offset: 0
O.g.f.: A(x) = 1 + x^2 + x^3 + 4*x^4 + 13*x^5 + 46*x^6 + 181*x^7 +...
where
A(x) = exp(-x) + x/(1-x)*exp(-x/(1-x)) + x^2/(1-2*x)^2/2!*exp(-x/(1-2*x)) + x^3/(1-3*x)^3/3!*exp(-x/(1-3*x)) + x^4/(1-4*x)^4/4!*exp(-x/(1-4*x)) + x^5/(1-5*x)^5/5!*exp(-x/(1-5*x)) + x^6/(1-6*x)^6/6!*exp(-x/(1-6*x)) +...
simplifies to a power series in x with integer coefficients.
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{a(n)=local(A=1+x,X=x+x*O(x^n));A=sum(k=0,n,1/(1-k*X)^k*x^k/k!*exp(-X/(1-k*X)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
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/* From a(n) = Sum_{k=1..n} Stirling2(n-k, k) * C(-1, k-1) */
{Stirling2(n, k) = sum(j=0, k, (-1)^(k+j) * binomial(k, j) * j^n) / k!}
{a(n)=if(n==0, 1, sum(k=1, n, Stirling2(n-k, k) * binomial(n-1, k-1)))}
for(n=0, 30, print1(a(n), ", "))
A245109
G.f.: Sum_{n>=0} exp(-(1 + n^2*x)) * (1 + n^2*x)^n / n!.
Original entry on oeis.org
1, 3, 31, 520, 11991, 350889, 12428746, 516450792, 24619176153, 1323971052261, 79280864647205, 5231080689880500, 377062508515478306, 29479066783583059530, 2484534527715953700780, 224559818606249783480400, 21666961097367611148157815, 2222844864226101120054773295
Offset: 0
G.f.: A(x) = 1 + 3*x + 31*x^2 + 520*x^3 + 11991*x^4 + 350889*x^5 +...
where
A(x) = exp(-1) + exp(-(1+x))*(1+x) + exp(-(1+2^2*x))*(1+2^2*x)^2/2!
+ exp(-(1+3^2*x))*(1+3^2*x)^3/3! + exp(-(1+4^2*x))*(1+4^2*x)^4/4!
+ exp(-(1+5^2*x))*(1+5^2*x)^5/5! + exp(-(1+6^2*x))*(1+6^2*x)^6/6!
+ exp(-(1+7^2*x))*(1+7^2*x)^7/7! + exp(-(1+8^2*x))*(1+8^2*x)^8/8! +...
simplifies to a power series in x with integer coefficients.
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Table[SeriesCoefficient[Sum[E^(-(1+k^2*x))*(1+k^2*x)^k/k!,{k,0,Infinity}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 12 2014 *)
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/* Must first set suitable precision */ \p300
{a(n)=local(A=1+x); A=suminf(k=0, exp(-(1+k^2*x)+x*O(x^n))*(1+k^2*x)^k/k!); round(polcoeff(A, n))}
for(n=0, 30, print1(a(n), ", "))
A245111
G.f.: A(x,y) = Sum_{n>=0} exp(-y/(1-n*x)) * 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
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.
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/* 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(""))
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/* 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(""))
Showing 1-3 of 3 results.
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