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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!. 3

%I

%S 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,

%T 5460,3150,462,0,1,504,10836,42000,46200,16632,1716,0,1,1143,43470,

%U 280665,519750,342342,84084,6435,0,1,2550,166375,1709400,4969965,5297292,2312310,411840,24310

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

%C Compare g.f. to: 1/(1-x*y) = Sum_{n>=0} exp(-y*(1+n*x)) * y^n*(1+n*x)^n / n!.

%C Row sums equal A245110.

%C Antidiagonal sums: A218667.

%C Main diagonal is: C(2*n-1,n) (A001700).

%C Secondary diagonal: C(2*n-1,n)*n^2 (A002544).

%H Paul D. Hanna, <a href="/A245111/b245111.txt">Table of n, a(n), of flattened triangle for rows 0..32</a>

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

%e G.f.: A(x,y) = 1 + x*y + x^2*(y + 3*y^2)

%e + x^3*(y + 12*y^2 + 10*y^3)

%e + x^4*(y + 35*y^2 + 90*y^3 + 35*y^4)

%e + x^5*(y + 90*y^2 + 525*y^3 + 560*y^4 + 126*y^5)

%e + x^6*(y + 217*y^2 + 2520*y^3 + 5460*y^4 + 3150*y^5 + 462*y^6) +...

%e where

%e A(x,y) = exp(-y) + exp(-y/(1-x))*y/(1-x) + (exp(-y/(1-2*x))*y^2/(1-2*x)^2)/2!

%e + (exp(-y/(1-3*x))*y^3/(1-3*x)^3)/3! + (exp(-y/(1-4*x))*y^4/(1-4*x)^4)/4!

%e + (exp(-y/(1-5*x))*y^5/(1-5*x)^5)/5! + (exp(-y/(1-6*x))*y^6/(1-6*x)^6)/6!

%e + (exp(-y/(1-7*x))*y^7/(1-7*x)^7)/7! + (exp(-y/(1-8*x))*y^8/(1-8*x)^8)/8! +...

%e simplifies to a power series with only integer coefficients of x^n*y^k.

%e Triangle begins:

%e 1;

%e 0, 1;

%e 0, 1, 3;

%e 0, 1, 12, 10;

%e 0, 1, 35, 90, 35;

%e 0, 1, 90, 525, 560, 126;

%e 0, 1, 217, 2520, 5460, 3150, 462;

%e 0, 1, 504, 10836, 42000, 46200, 16632, 1716;

%e 0, 1, 1143, 43470, 280665, 519750, 342342, 84084, 6435;

%e 0, 1, 2550, 166375, 1709400, 4969965, 5297292, 2312310, 411840, 24310;

%e 0, 1, 5621, 615780, 9754030, 42567525, 68549481, 47087040, 14586000, 1969110, 92378; ...

%e where T(n,k) = A048993(n,k) * C(n+k-1, k-1) for k>0.

%o (PARI) /* From definition: */

%o {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)}

%o for(n=0, 10, for(k=0,n, print1(T(n,k),", "));print(""))

%o (PARI) /* From T(n,k) = Stirling2(n, k) * C(n+k-1, k-1) */

%o {Stirling2(n, k) = sum(j=0, k, (-1)^(k+j) * binomial(k, j) * j^n) / k!}

%o {T(n,k)=if(k==0,0^n,Stirling2(n, k) * binomial(n+k-1, k-1))}

%o for(n=0, 10, for(k=0,n, print1(T(n,k),", "));print(""))

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

%K nonn,tabl

%O 0,6

%A _Paul D. Hanna_, Jul 12 2014

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Last modified January 23 03:15 EST 2022. Contains 350504 sequences. (Running on oeis4.)