A176277 Sum over the odd entries of the rows in the triangle Worpitzky(n, k)*Harmonic(k) (A176276).

0, 1, 3, 18, 125, 1020, 9667, 104790, 1281177, 17457840, 262493231, 4318429962, 77178551749, 1489209086820, 30859393432155, 683549418431934, 16118484827641841, 403156528379483160, 10661349675027656839

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..420

Peter Luschny, A sequence transformation and the Bernoulli numbers.

Formula

a(n) = Sum_{k=0..n} (k mod 2) abs(Stirling1(k+1, 2)*Stirling2(n+1, k+1)).

Example

Let W(n, k) be the Worpitzky numbers and H(n) the harmonic numbers.
a(3) = W(3,1)H(1) + W(3,3)H(3) = 7*1 + 6*(11/6) = 18.

Maple

A176277 := proc(n) local k; add((k mod 2)*T176276(n, k), k=0..n) end;

Mathematica

a[1] = 1; a[n_]:= Sum[ StirlingS2[n+1, k+1]*k!*HarmonicNumber[k], {k, 0, n, 2}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 30 2013 *)

Prog

(PARI) a(n) = if(n<2, n, sum(k=0, n, k!*stirling(n+1, k+1, 2)*sum(j=1, k, 1/j)) ); \\ G. C. Greubel, Nov 24 2019
(Magma) [n lt 2 select n else (&+[Abs(StirlingFirst(k+1, 2)*StirlingSecond(n+1, k+1)): k in [0..n]])/2: n in [0..25]];
(Sage)
def a(n):
    if (n<2): return n
    else: return sum( factorial(k)*stirling_number1(n+1, k+1)*harmonic_number(k) for k in (0..n))/2
[a(n) for n in (0..25)] # G. C. Greubel, Nov 24 2019
(GAP)
a:= function(n)
    if n<2 then return n;
    else return Sum([0..n], k-> AbsInt(Stirling1(k+1, 2) * Stirling2(n+1, k+1)))/2;
    fi; end;
List([0..25], n-> a(n)); # G. C. Greubel, Nov 24 2019

Crossrefs

Cf. A028246, A176276.

Sequence in context: A181998 A365130 A349024 * A289429 A004987 A340894

Adjacent sequences: A176274 A176275 A176276 * A176278 A176279 A176280

Keyword

easy,nonn

Author

Peter Luschny, Apr 14 2010

Status

approved