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A001701 Generalized Stirling numbers.
(Formerly M4169 N1735)
19
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

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.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

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

MAPLE

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

MATHEMATICA

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 *)

From Robert A. Russell, Oct 04 2018: (Start)

Prepend[Table[Coefficient[Product[x+j, {j, 2, k}], x, k-3], {k, 3, 40}], 1]

Prepend[LinearRecurrence[{5, -10, 10, -5, 1}, {6, 26, 71, 155, 295}, 40], 1] (End)

PROG

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

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

(PARI) 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

CROSSREFS

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

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

Sequence in context: A190095 A135036 A166796 * A241452 A175898 A255870

Adjacent sequences:  A001698 A001699 A001700 * A001702 A001703 A001704

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 19 22:00 EDT 2018. Contains 316378 sequences. (Running on oeis4.)