A191490 Triangle generated by the recurrence T(n+1,k+1) = T(n,k+1) + n * T(n,k) + delta(n,k) with the initial values T(n,0) = 1 and T(0,k) = delta(k,0), where delta(n,k) is the Kronecker delta.

1, 1, 1, 1, 2, 2, 1, 4, 6, 5, 1, 7, 18, 23, 16, 1, 11, 46, 95, 108, 65, 1, 16, 101, 325, 583, 605, 326, 1, 22, 197, 931, 2533, 4103, 3956, 1957, 1, 29, 351, 2310, 9050, 21834, 32677, 29649, 13700, 1, 37, 583, 5118, 27530, 94234, 207349, 291065, 250892, 109601, 1, 46, 916, 10365, 73592, 342004, 1055455, 2157206, 2870477, 2367629, 986410

Offset

0,5

Comments

Row sums = A000522.

Diagonal sums = A191491.

Central coefficients = A191492.

Binomial row sums = A191493.

Let r(n) = sum(T(n,k),k=0..n) be the row sums.

Let s(n) = sum(T(n,k)*(-1)^(n-k),k=0..n) be the alternated row sums.

Let d(n) = T(n,n) be the diagonal elements.

Then s(n+2) = r(n) and r(n) = d(n+1).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..560

Formula

Recurrence: T(n+1,k+1) = sum(i*T(i,k),i=0..n)+[k<=n],

where [k<=n]=1 if k<=n and [k<=n]=0 if k>n.

Mixed generating series:

sum(T(n,k)*q^k*x^n/n!,n=0..inf) = (1-q*x)^(-1/q)*(1+q*int(exp(q*t)/(1-q*t)^((q-1)/q),t=0..x)).

Let f(n,q)= sum(T(n,k)*q^k,k=0..n) the generating polynomials of the rows. Then f(n+1,q)=(1+n*q)*f(n,q)+q^(n+1).

Let A(n,q)=sum(s(n,n-k)*q^k,k=0..n), where the coefficients s(n,k) are the (signless) Stirling numbers of the first kind.

Let B(n,q)=sum(sum(binomial(n-1,i)*s(n-i-1,k),i=0..n-1)*(q-1)^k*q^(n-k),k=0..n-1). Finally, let P(n,q)=A(n,q)+sum(binomial(n,k)*A(k,q)*B(n-k,q),k=0..n). Then T(n,k)=[q^k]P(n,q).

Example

Triangle begins:
1
1, 1
1, 2, 2
1, 4, 6, 5
1, 7, 18, 23, 16
1, 11, 46, 95, 108, 65
1, 16, 101, 325, 583, 605, 326
1, 22, 197, 931, 2533, 4103, 3956, 1957
1, 29, 351, 2310, 9050, 21834, 32677, 29649, 13700

Mathematica

f[n_, k_] := f[n, k] = f[n - 1, k] + (n - 1)f[n - 1, k - 1] + If[n == k, 1, 0]
f[_, 0] = 1;
f[0, _] = 0;
Flatten[Table[f[n, k], {n, 0, 100}, {k, 0, n}]]

Prog

(Maxima) P[0]:1$
P[n]:=(1+(n-1)*q)*P[n-1]+q^n$
create_list(coeff(expand(P[n]), q^k), n, 0, 12, k, 0, n);

Crossrefs

Cf. A000522, A191491, A191492, A191493.

Sequence in context: A369632 A363493 A193597 * A061598 A328873 A071946

Adjacent sequences: A191487 A191488 A191489 * A191491 A191492 A191493

Keyword

nonn,tabl

Author

Emanuele Munarini, Jun 03 2011

Status

approved