|
| |
|
|
A074051
|
|
a(n) = amount of Sum_{i=1..m} (i+1)! in Sum_{i=1..m} i^n*(i+1)!.
|
|
8
| |
|
|
1, -1, 0, 3, -7, 0, 59, -217, 146, 2593, -15551, 32802, 160709, -1856621, 7971872, 1299951, -287113779, 2262481448, -7275903849, -36989148757, 698330745002, -4867040141851, 10231044332629, 184216198044034, -2679722886596295, 17971204188130391, -17976259717948832
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,4
|
|
|
COMMENTS
| If a(n)=0 then Sum_{i=1..infty}i^n(i+1)! = b(n) in the p-adic numbers. The only known numbers n with a(n)=0 are 2 and 5.
a(n)*(-1)^n gives the alternating row sums of the Sheffer triangle A143494 (2-restricted Stirling2). - From Wolfdieter Lang, Oct 06 2011
|
|
|
FORMULA
| For each n there uniquely determined numbers a(n) and b(n) and a polynomial p_n such that for all integers m: Sum_{i=1..m}i^n(i+1)! = a(n)*Sum_{i=1..m}(i+1)! + p_n(m)(m+2)! + b(n) The sequence b(n) is A074052.
Second inverse binomial transform of A000587. E.g.f.: exp(1-2*x-exp(-x)). G.f.: Sum((x/(1+2*x))^k/Product(1+l*x/(1+2*x), l = 0 .. k), k = 0 .. infinity)/(1+2*x). a(n) = Sum_{k=0..n} (-1)^(n-k)*(k^2-3*k+1)*Stirling2(n, k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 27 2005
a(n) = (-1)^n*(A000587(n+2)-A000587(n+1)). -Peter Luschny, Apr 17 2011
|
|
|
EXAMPLE
| a(2)=0 because Sum_{i=1..m}i^2(i+1)! = (m-1)(m+2)!+2. a(3)=3 because Sum_{i=1..m}i^3(i+1)! = 3*Sum_{i=1..m}(i+1)!+(m^2-m-1)(m+2)!+2.
|
|
|
MAPLE
| alias(S2 = combinat[stirling2]);
A074051 := proc(n) local k;
1 + add((-1)^(n+k) * (S2(n+1, k+1) - S2(n+2, k+1)), k = 0..n) end:
seq(A074051(i), i = 0..26); # - Peter Luschny, Apr 17 2011
|
|
|
MATHEMATICA
| A[a_] := Module[{p, k}, p[n_] = 0; For[k = a - 1, k >= 0, k--, p[n_] = Expand[p[n] + n^k Coefficient[n^a - (n + 2)p[n] + p[n - 1], n^(k + 1)]] ]; Expand[n^a - (n + 2)p[n] + p[n - 1]] ]
|
|
|
CROSSREFS
| Cf. A074052, A143494.
Sequence in context: A103844 A199068 A198490 * A048292 A072450 A085785
Adjacent sequences: A074048 A074049 A074050 * A074052 A074053 A074054
|
|
|
KEYWORD
| easy,sign
|
|
|
AUTHOR
| Jan Fricke (fricke(AT)uni-greifswald.de), Aug 14 2002
|
|
|
EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 27 2005
|
| |
|
|
