A111885 Row sums of triangle A112492.

1, 2, 5, 20, 152, 2542, 100326, 10194844, 2809233510, 2212797607312, 5359196565766782, 39928779843430949176, 1018129474625651322506886, 85890171235256453902613870992, 26477529277143069417959927152215342

Offset: 0

Author

Wolfdieter Lang, Sep 12 2005

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..55

Crossrefs

Cf. A112492.

Programs

Magma

T:= func< n,k | (-1)*Factorial(k+1)^(n-k)*(&+[(-1)^j*Binomial(k+1,j)/j^(n-k) : j in [1..k+1]]) >; // T = A112492
A111885:= func< n | (&+[T(n,k): k in [0..n]]) >;
[A111885(n): n in [0..40]]; // G. C. Greubel, Jul 24 2023

Mathematica

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, (k+1)^(n-k)*T[n-1, k-1] + k!*T[n-1, k]];
a[n_]:= a[n]= Sum[T[n,k], {k,0,n}]; (* T = A112492 *)
Table[a[n], {n,0,40}] (* G. C. Greubel, Jul 24 2023 *)

SageMath

@CachedFunction
def T(n,k): # T = A112492
    if (k==0 or k==n): return 1
    else: return (k+1)^(n-k)*T(n-1,k-1) + factorial(k)*T(n-1,k)
def A111885(n): return sum(T(n,k) for k in range(n+1))
[A111885(n) for n in range(31)] # G. C. Greubel, Jul 24 2023

Formula

a(n) = Sum_{j=0..n} A112492(n, j), n >= 0.