A074052 -id:A074052 - cp's OEIS frontend

A074051 For each n there are uniquely determined numbers a(n) and b(n) and a polynomial p_n(x) such that for all integers m we have 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.

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

Offset: 0

Jan Fricke, Aug 14 2002

If a(n)=0 then Sum_{i>=1}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). - Wolfdieter Lang, Oct 06 2011

			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.

Robert Israel, Table of n, a(n) for n = 0..545

Cf. A074052, A143494.

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

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]] ]
(* Second program: *)
a[n_] := (-1)^n (BellB[n+2, -1] - BellB[n+1, -1]);
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 21 2018, after Peter Luschny *)

from itertools import accumulate
def A074051_list(size):
    if size < 1: return []
    L, accu = [], [1]
    for n in range(size-1):
        accu = list(accumulate([-accu[-1]] + accu))
        L.append(-(-1)**n*accu[-2])
    return L
print(A074051_list(28)) # Peter Luschny, Apr 25 2016

From Vladeta Jovovic, Jan 27 2005: (Start)
Second inverse binomial transform of A000587.
E.g.f.: exp(1 - 2*x - exp(-x)).
G.f.: Sum_{k >= 0}((x/(1+2*x))^k/Product_{l=0..k}(1 + l*x/(1+2*x)))/(1+2*x).
a(n) = Sum_{k=0..n} (-1)^(n-k)*(k^2-3*k+1)*Stirling2(n, k). (End)
a(n) = (-1)^n*(A000587(n+2)-A000587(n+1)). - Peter Luschny, Apr 17 2011
From Sergei N. Gladkovskii, Sep 28 2012 to Apr 22 2013: (Start)
Continued fractions:
G.f.: 1/U(0) where U(k)= x*k + 1 + x + x^2*(k+1)/U(k+1).
G.f.: -1/U(0) where U(k)= -x*k - 1 - x + x^2*(k+1)/U(k+1).
G.f.: 1/(U(0) - x) where U(k)= 1 + x + x*(k+1)/(1 - x/U(k+1)).
G.f.: 1/(U(0) + x) where U(k)= 1 + x*(2*k+1) - x*(k+1)/(1 + x/U(k+1)).
G.f.: 1/G(0) where G(k)= 1 + 2*x/(1 + 1/(1 + 2*x*(k+1)/G(k+1))).
G.f.: 1 - 2*x/(G(0) + 2*x) where G(k)= 1 + 1/(1 + 2*x*(k+1)/(1 + 2*x/G(k+1))).
G.f.: (G(0) - 1)/(x-1) where G(k) =  1 - 1/(1+k*x+2*x)/(1-x/(x-1/G(k+1))).
G.f.: (G(0)-2-2*x)/x^2 where G(k) = 1 + 1/(1+k*x)/(1-x/(x+1/G(k+1) )).
G.f.: (S-2-2*x)/x^2 where S = sum(k>=0, (2 + x*k)*x^k/prod(i=0..k, (1+x*i))).
G.f.: (G(0)-2)/x where G(k) = 1 + 1/(1+k*x+x)/(1-x/(x+1/G(k+1))).
G.f.: (1+x)/x/Q(0) - 1/x, where Q(k)=  1 + x - x/(1 + x*(k+1)/Q(k+1)). (End)
a(n) = exp(1) * (-1)^n * Sum_{k>=0} (-1)^k * (k+2)^n / k!. - Ilya Gutkovskiy, Sep 02 2021

More terms from Vladeta Jovovic, Jan 27 2005

A197184 Triangle of polynomial coefficients of the polynomial factors defined in A074051.

Original entry on oeis.org

1, -1, 1, -1, -1, 1, 7, -2, -1, 1, -13, 12, -3, -1, 1, -17, -22, 18, -4, -1, 1, 199, -45, -35, 25, -5, -1, 1, -605, 465, -84, -53, 33, -6, -1, 1, -225, -1449, 910, -133, -77, 42, -7, -1, 1, 11703, -864, -3094, 1594, -190, -108, 52, -8, -1, 1, -59317, 33780, -1380, -6027, 2583, -252, -147, 63, -9, -1, 1, 83143, -179398, 78567, -771, -10899, 3948, -315, -195, 75, -10, -1, 1, 991671, 271073, -461978, 159115, 2882, -18546, 5764, -374, -253, 88, -11, -1, 1

Offset: 1

R. J. Mathar, Oct 11 2011

The triangle T(n,k), 0<=kA074052(n) + A074051(n)*sum_{i=1..m} (i+1)! + p_n(m) *(m+2)!.

			1;   1
-1,1;  -1+x
-1,-1,1;  -1-x+x^2
7,-2,-1,1;  7-2*x-x^2+x^3
-13,12,-3,-1,1;  -13+12*x-3*x^2-x^3+x^4
-17,-22,18,-4,-1,1;   -17-22*x+18*x^2-4*x^3-x^4+x^5

A074052(n) + 2*A074051(n) + 6*p_n(1) = 2. - R. J. Mathar, Oct 13 2011
(x+2)*p_n(x)-p_n(x-1) = x^n-A074051(n). - R. J. Mathar, Oct 13 2011
Conjectures on p_n(x)= sum_{k=0..n-1} T(n,k)*x^k:
T(n,n-1) = 1.
T(n,n-2) = -1.
T(n,n-3) = -(n-2).
T(n,n-4) = A055998(n-2).
T(n,n-5) = -(n-2)*(n^2-4*n+21)/6.
T(n,n-6) = (n-5)*(n-2)*(n^2-19*n-24)/24.

cp's OEIS Frontend

A074051 For each n there are uniquely determined numbers a(n) and b(n) and a polynomial p_n(x) such that for all integers m we have 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.

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A197184 Triangle of polynomial coefficients of the polynomial factors defined in A074051.

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cp's OEIS Frontend

A074051 For each n there are uniquely determined numbers a(n) and b(n) and a polynomial p_n(x) such that for all integers m we have 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.

Views

Author

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Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A197184 Triangle of polynomial coefficients of the polynomial factors defined in A074051.

Views

Author

Keywords

Comments

Examples

Formula

A074051 For each n there are uniquely determined numbers a(n) and b(n) and a polynomial p_n(x) such that for all integers m we have 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.