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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums = A000522.

Diagonal sums = A191491.

Central coefficients = A191492.

Binomial row sums = A191493.

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

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).

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: A228336 A111062 A193597 * A061598 A328873 A071946

Adjacent sequences:  A191487 A191488 A191489 * A191491 A191492 A191493

KEYWORD

nonn,tabl

AUTHOR

Emanuele Munarini, Jun 03 2011

STATUS

approved

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Last modified March 3 14:58 EST 2021. Contains 341762 sequences. (Running on oeis4.)