A059302 -id:A059302 - cp's OEIS frontend

A001701 Generalized Stirling numbers.

1, 6, 26, 71, 155, 295, 511, 826, 1266, 1860, 2640, 3641, 4901, 6461, 8365, 10660, 13396, 16626, 20406, 24795, 29855, 35651, 42251, 49726, 58150, 67600, 78156, 89901, 102921, 117305, 133145, 150536, 169576, 190366, 213010, 237615, 264291, 293151, 324311

Offset: 1

N. J. A. Sloane

For n>3, a(n-2) gives the number of bounded regions created when the pairwise perpendicular bisectors of n points divide the Euclidean plane into a maximum of A308305(n) regions. This is also equivalent to the number of regions lost from A308305(n) when n>3 points move from maximal position to a circle. - Alvaro Carbonero, Elizabeth Castellano, Charles Kulick, Karie Schmitz, Jul 26 2019

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Alvaro Carbonero, Beth Anne Castellano, Gary Gordon, Charles Kulick, Karie Schmitz, and Brittany Shelton, Permutations of point sets in R_d, arXiv:2106.14140 [math.CO], 2021.
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Equals A059302(n+2) + 1, n>1. Partial sums of A005564.

For n>1, a(n) = A145324(n+1,3).

Concatenation([1],List([2..40],n->n*(n-1)*(3*n^2+17*n+26)/24)); # Muniru A Asiru, Sep 29 2018

[1] cat [n*(n-1)*(3*n^2 + 17*n + 26)/24: n in [2..40]]; // Vincenzo Librandi, Sep 30 2018

A001701 := proc(n)
    if n = 1 then
        1;
    else
        n*(n-1)*(3*n^2+17*n+26)/24 ;
    end if;
end proc: # R. J. Mathar, Sep 23 2016

f[k_] := k + 1; t[n_] := Table[f[k], {k, 1, n}]; a[n_] := SymmetricPolynomial[2, t[n]]; Join[{1}, Table[a[n], {n, 2, 30}]] (* Clark Kimberling, Dec 31 2011 *)
Join[{1}, Table[n (n - 1) (3 n^2 + 17 n + 26) / 24, {n, 2, 40}]] (* Vincenzo Librandi, Sep 30 2018 *)
CoefficientList[Series[(-1 - x - 6 x^2 + 9 x^3 - 5 x^4 + x^5)/(-1 + x)^5, {x, 0, 30}], x] (* Stefano Spezia, Sep 30 2018 *)
Prepend[Table[Coefficient[Product[x+j, {j,2,k}], x, k-3], {k,3,40}],1] (* or *) Prepend[LinearRecurrence[{5, -10, 10, -5, 1}, {6, 26, 71, 155, 295}, 40],1] (*Robert A. Russell, Oct 04 2018 *)

Vec(x*(-1-x-6*x^2+9*x^3-5*x^4+x^5)/(-1+x)^5+O(x^30)) \\ Stefano Spezia, Sep 30 2018

a(n) = n*(n-1)*(3n^2 + 17n + 26)/24, n > 1.
G.f.: z*(-1-z-6*z^2+9*z^3-5*z^4+z^5)/(z-1)^5. - Simon Plouffe in his 1992 dissertation
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i) * Product_{j=0..k-1} (-a - j), then a(n) = f(n,n-2,2), for n >= 2. - Milan Janjic, Dec 20 2008
For n>1, a(n) = A308305(n+2) - (n^2 + 3n + 2). - Alvaro Carbonero, Elizabeth Castellano, Charles Kulick, Karie Schmitz, Jul 20 2019
E.g.f.: x + (1/24)*exp(x)*x^2*(72 + 32*x + 3*x^2). - Stefano Spezia, Sep 07 2019
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Colin Barker, Jul 08 2020

A008296 Triangle of Lehmer-Comtet numbers of the first kind.

Original entry on oeis.org

1, 1, 1, -1, 3, 1, 2, -1, 6, 1, -6, 0, 5, 10, 1, 24, 4, -15, 25, 15, 1, -120, -28, 49, -35, 70, 21, 1, 720, 188, -196, 49, 0, 154, 28, 1, -5040, -1368, 944, 0, -231, 252, 294, 36, 1, 40320, 11016, -5340, -820, 1365, -987, 1050, 510, 45, 1, -362880, -98208, 34716, 9020, -7645, 3003, -1617, 2970, 825, 55, 1, 3628800

Offset: 1

N. J. A. Sloane

Triangle arising in the expansion of ((1+x)*log(1+x))^n.

Also the Bell transform of (-1)^(n-1)*(n-1)! if n>1 else 1 adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 16 2016

			Triangle begins:
   1;
   1,  1;
  -1,  3,   1;
   2, -1,   6,  1;
  -6,  0,   5, 10,  1;
  24,  4, -15, 25, 15, 1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139.

Alois P. Heinz, Rows n = 1..141, flattened
H. W. Gould, A Set of Polynomials Associated with the Higher Derivatives of y = x^x, Rocky Mountain J. Math. Volume 26, Number 2 (1996), 615-625.
Tian-Xiao He and Yuanziyi Zhang, Centralizers of the Riordan Group, arXiv:2105.07262 [math.CO], 2021.
D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.

Cf. A039621 (second kind), A354795 (variant), A185164, A005727 (row sums), A298511 (central).
Columns: A045406 (column 2), A347276 (column 3), A345651 (column 4).
Diagonals: A000142, A000217, A059302.
Cf. A176118.

for n from 1 to 20 do for k from 1 to n do
printf(`%d,`, add(binomial(l,k)*k^(l-k)*Stirling1(n,l), l=k..n)) od: od:
# second program:
A008296 := proc(n, k) option remember; if k=1 and n>1 then (-1)^n*(n-2)! elif n=k then 1 else (n-1)*procname(n-2, k-1) + (k-n+1)*procname(n-1, k) + procname(n-1, k-1) end if end proc:
seq(print(seq(A008296(n, k), k=1..n)), n=1..7); # Mélika Tebni, Aug 22 2021

a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n-2)!;
a[n_, n_] = 1; a[n_, k_] := a[n, k] = (n-1) a[n-2, k-1] + a[n-1, k-1] + (k-n+1) a[n-1,k]; Flatten[Table[a[n, k], {n, 1, 12}, {k, 1, n}]][[1 ;; 67]]
(* Jean-François Alcover, Apr 29 2011 *)

{T(n, k) = if( k<1 || k>n, 0, n! * polcoeff(((1 + x) * log(1 + x + x * O(x^n)))^k / k!, n))}; /* Michael Somos, Nov 15 2002 */

# uses[bell_matrix from A264428]
# Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.
bell_matrix(lambda n: (-1)^(n-1)*factorial(n-1) if n>1 else 1, 7) # Peter Luschny, Jan 16 2016

E.g.f. for a(n, k): (1/k!)[ (1+x)*log(1+x) ]^k. - Len Smiley
Left edge is (-1)*n!, for n >= 2. Right edge is all 1's.
a(n+1, k) = n*a(n-1, k-1) + a(n, k-1) + (k-n)*a(n, k).
a(n, k) = Sum_{m} binomial(m, k)*k^(m-k)*Stirling1(n, m).
From Peter Bala, Mar 14 2012: (Start)
E.g.f.: exp(t*(1 + x)*log(1 + x)) = Sum_{n>=0} R(n,t)*x^n/n! = 1 + t*x + (t+t^2)x^2/2! + (-t+3*t^2+t^3)x^3/3! + .... Cf. A185164. The row polynomials R(n,t) are of binomial type and satisfy the recurrence R(n+1,t) = (t-n)*R(n,t) + t*d/dt(R(n,t)) + n*t*R(n-1,t) with R(0,t) = 1 and R(1,t) = t. Inverse array is A039621.
(End)
Sum_{k=0..n} (-1)^k * a(n,k) = A176118(n). - Alois P. Heinz, Aug 25 2021

More terms from James Sellers, Jan 26 2001

Edited by N. J. A. Sloane at the suggestion of Andrew Robbins, Dec 11 2007

A241765 a(n) = n(n + 1)(n + 2)(3n + 17)/24.

Original entry on oeis.org

0, 5, 23, 65, 145, 280, 490, 798, 1230, 1815, 2585, 3575, 4823, 6370, 8260, 10540, 13260, 16473, 20235, 24605, 29645, 35420, 41998, 49450, 57850, 67275, 77805, 89523, 102515, 116870, 132680, 150040, 169048, 189805, 212415, 236985, 263625, 292448

Offset: 0

Bruno Berselli, Apr 28 2014

Equivalently, Sum_{i=0..n} (i+4)*A000217(i).
Sequences of the type Sum_{i=0..n} (i+k)*A000217(i):
k = 0,  A001296: 0,  1,  7, 25,  65, 140, 266, 462, ...
k = 1,  A000914: 0,  2, 11, 35,  85, 175, 322, 546, ...
k = 2,  A050534: 0,  3, 15, 45, 105, 210, 378, 630, ... (deleting two 0)
k = 3,  A215862: 0,  4, 19, 55, 125, 245, 434, 714, ...
k = 4,     a(n): 0,  5, 23, 65, 145, 280, 490, 798, ...
k = 5,  A239568: 0,  6, 27, 75, 165, 315, 546, 882, ...
Antidiagonal sums (without 0) give A034263: 1, 9, 39, 119, 294, ...
Diagonal: 1, 11, 45, 125, 280, 546, ... is A051740.
Also: k = -1 gives A050534 deleting a 0; k = -2 gives 0 followed by A059302.
After 0, partial sums of A212343 and third column of A118788.
This sequence is even related to A005286 by a(n) = n*A005286(n) - Sum_{i=0..n-1} A005286(i).

			a(7) = 4*0 + 5*1 + 6*3 + 7*6 + 8*10 + 9*15 + 10*21 + 11*28 = 798.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A005286, A118788, A212343, A227342.

Cf. similar sequences A000914, A001296, A050534, A059302, A215862, A239568 (see table in Comments lines).

/* By first comment: */ k:=4; A000217:=func; [&+[(i+k)*A000217(i): i in [0..n]]: n in [0..40]];

Maple

A241765:=n->n*(n + 1)*(n + 2)*(3*n + 17)/24; seq(A241765(n), n=0..40); # Wesley Ivan Hurt, May 09 2014

Mathematica

Table[n (n + 1) (n + 2) (3 n + 17)/24, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 23, 65, 145}, 40] CoefficientList[Series[x (5 - 2 x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 09 2014 *)

Maxima

makelist(coeff(taylor(x*(5-2*x)/(1-x)^5, x, 0, n), x, n), n, 0, 40);

PARI

a(n)=n*(n+1)*(n+2)*(3*n+17)/24 \\ Charles R Greathouse IV, Oct 07 2015

PARI

x='x+O('x^99); concat(0, Vec(x*(5-2*x)/(1-x)^5)) \\ Altug Alkan, Apr 10 2016

Sage

[n*(n+1)*(n+2)*(3*n+17)/24 for n in (0..40)]

Formula

G.f.: x*(5 - 2*x)/(1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

a(n) = A227342(A055998(n+1)).

a(n) = Sum_{j=0..n+2} (-1)^(n-j)*binomial(-j,-n-2)*S1(j,n), S1 Stirling cycle numbers A132393. - Peter Luschny, Apr 10 2016

cp's OEIS Frontend

A001701 Generalized Stirling numbers.

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A008296 Triangle of Lehmer-Comtet numbers of the first kind.

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A241765 a(n) = n*(n + 1)*(n + 2)*(3*n + 17)/24.

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A193002 Triangle T(n,k)=0 (k odd), T(0,0)=-3, T(n,0)=1 (n > 0) and T(n,k) = T(n-1,k) - T(n-2,k-2).

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A241765 a(n) = n(n + 1)(n + 2)(3n + 17)/24.