A163931 -id:A163931 - cp's OEIS frontend

A001722 Generalized Stirling numbers.

1, 18, 251, 3325, 44524, 617624, 8969148, 136954044, 2201931576, 37272482280, 663644774880, 12413008539360, 243533741849280, 5003753991174720, 107497490419296000, 2410964056571616000, 56366432074677312000, 1371711629236971456000, 34699437370290760704000

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=3,n=5) ~ exp(-x)/x^3*(1 - 18/x + 251/x^2 - 3325/x^3 + 44524/x^4 - 617624/x^5 + ... ) leads to the sequence given above. See A163931 and A163932 for more information. - Johannes W. Meijer, Oct 20 2009

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 = 0..100
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

Table[Sum[(-1)^(n + k)*Binomial[k + 2, 2]*5^k*StirlingS1[n + 2, k + 2], {k, 0, n}], {n, 0, 20}] (* T. D. Noe, Aug 10 2012 *)

a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+2, 2)*5^k*Stirling1(n+2, k+2). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

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-2) = |f(n,2,5)|, for n >= 2. - Milan Janjic, Dec 21 2008

More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

A001723 Generalized Stirling numbers.

Original entry on oeis.org

1, 26, 485, 8175, 134449, 2231012, 37972304, 668566300, 12230426076, 232959299496, 4623952866312, 95644160132976, 2060772784375824, 46219209678691200, 1078100893671811200, 26129183717351462400, 657337657573760947200, 17147815411007234188800

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=4,n=5) ~ exp(-x)/x^4*(1 - 26/x + 485/x^2 - 8175/x^3 + 134449/x^4 - 2231012/x^5 + ...) leads to the sequence given above. See A163931 and A163934 for more information. - Johannes W. Meijer, Oct 20 2009

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 = 0..100
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

Table[Sum[(-1)^(n + k)*Binomial[k + 3, 3]*5^k*StirlingS1[n + 3, k + 3], {k, 0, n}], {n, 0, 20}] (* T. D. Noe, Aug 10 2012 *)

a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(3+k, 3)*5^k*Stirling1(n+3, k+3). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

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-3) = |f(n,3,5)|, for n >= 3. - Milan Janjic, Dec 21 2008

More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

A001724 Generalized Stirling numbers.

Original entry on oeis.org

1, 35, 835, 17360, 342769, 6687009, 131590430, 2642422750, 54509190076, 1159615530788, 25497032420496, 580087776122400, 13662528306823824, 333132304121991504, 8407011584355624288, 219490450157530821024, 5925108461354500651776, 165275526944869750483200

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=5,n=5) ~ exp(-x)/x^5*(1 - 35/x + 835/x^2 - 17360/x^3 + 342769/x^4 - ...) leads to the sequence given above. See A163931 for E(x,m,n) information and A163932 for a Maple procedure for the asymptotic expansion. - Johannes W. Meijer, Oct 20 2009

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 = 0..100
D. S. Mitrinovic, M. S. Mitrinovic, Tableaux d'une classe de nombres relies aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).
Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, arXiv version, arXiv:1103.3453 [math-ph], 2011.

Table[Sum[(-1)^(n + k)*Binomial[k + 4, 4]*5^k*StirlingS1[n + 4, k + 4], {k, 0, n}], {n, 0, 20}] (* T. D. Noe, Aug 10 2012 *)

a(n) = sum(k=0, n, (-1)^(n+k)*binomial(k+4, 4)*5^k*stirling(n+4, k+4, 1)) \\ Michel Marcus, Jan 20 2016

a(n) = sum((-1)^(n+k)*binomial(k+4, 4)*5^k*stirling1(n+4, k+4), k=0..n). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
E.g.f.: (6-156*log(1-x)+753*log(1-x)^2-1066*log(1-x)^3+420*log(1-x)^4)/(6*(1-x)^9). - Vladeta Jovovic, Mar 01 2004
If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n-4) = |f(n,4,5)|, for n>=4. - Milan Janjic, Dec 21 2008

More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

A262343 Numerator of 3*(1-2/n), for n >= 3.

Original entry on oeis.org

1, 3, 9, 2, 15, 9, 7, 12, 27, 5, 33, 18, 13, 21, 45, 8, 51, 27, 19, 30, 63, 11, 69, 36, 25, 39, 81, 14, 87, 45, 31, 48, 99, 17, 105, 54, 37, 57, 117, 20, 123, 63, 43, 66, 135, 23, 141, 72, 49, 75, 153, 26, 159, 81, 55, 84, 171, 29, 177, 90, 61, 93, 189, 32

Offset: 3

Kival Ngaokrajang, Sep 18 2015

Given a regular n-gon with side length s, draw a circular arc of radius s around each of the n-gon's vertices so as to connect that vertex's two nearest neighbors, drawing the arc on the shorter side of the circle; i.e., each arc will extend through an angle of Pi*(n-2)/n radians (see illustration). Connect the n arcs thus drawn into a single closed curve if n is odd, or into a pair of identical (but with one rotated by 2*Pi/n radians with respect to the other) overlapping closed curves if n is even. The arcs and the curve (or pair of curves) have the following properties:
(i) Since the length L(n) of each single arc is L(n) = s*Pi*(n-2)/n, the ratio of the length of a single arc for an n-gon to the length of a single arc for the n=3 case is L(n)/L(3) = (s*Pi*(n-2)/n)/(s*Pi*(3-2)/3) = 3(1-2/n). The numerator and denominator of 3(1-2/n) are a(n) and A060789(n) respectively.
(ii) Since the loop length (considering only one of the two loops when there are two overlapping loops) is L(n)*n when n is odd, or L(n)*n/2 when n is even, the ratio of the loop length for an n-gon to the loop length for the n=3 case is (L(n)*n)/(L(3)*3) = (s*Pi*(n-2))/(s*Pi) = n-2 when n is odd, or (L(n)*n/2)/(L(3)*3) = (s*Pi*(n-2)/2)/(s*Pi) = (n-2)/2 when n is even; thus, whether odd or even, that ratio is numerator(1-2/n) = A026741(n-2).
The moment generating function of p(x, m=1, n=2, mu=2) = 3*x*E(x, 1, 2), see A163931 and A274181, is given by M(a) = (3*a-6)/(a^2*(a-1)) + 6*log(1-a)/a^3. The series expansion of M(a) leads to the sequence given above. - Johannes W. Meijer, Jul 04 2016

Peter Kagey, Table of n, a(n) for n = 3..10000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Cf. A060789, A026741.

[Numerator(3*(1-2/n)): n in [3..80]]; // Vincenzo Librandi, Sep 19 2015

a:= proc(n): numer(3*(n-2)/n) end: seq(a(n), n=3..66); # Johannes W. Meijer, Jul 03 2016

Table[Numerator[3 (1 - 2/n)], {n, 3, 60}] (* Michael De Vlieger, Sep 18 2015 *)

{for(n=3, 100, a=numerator(3*(1-2/n)); print1 (a, ", "))}

Vec(x^3*(3*x^10+x^9+9*x^8+6*x^7+5*x^6+9*x^5+15*x^4+2*x^3+9*x^2+3*x+1)/((x-1)^2*(x+1)^2*(x^2-x+1)^2*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Sep 20 2015

a(n) = numerator(3*(1-2/n)), for n >= 3.
From Peter Kagey, Sep 18 2015: (Start)
For integers k:
a(6k+0) =  3 * k - 1
a(6k+1) = 18 * k - 3
a(6k+2) =  9 * k + 1
a(6k+3) =  6 * k + 1
a(6k+4) =  9 * k + 3
a(6k+5) = 18 * k + 9
(End)
From Colin Barker, Sep 20 2015: (Start)
a(n) = 2*a(n-6) - a(n-12).
G.f.: x^3*(3*x^10+x^9+9*x^8+6*x^7+5*x^6+9*x^5+15*x^4+2*x^3+9*x^2+3*x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)^2*(x^2+x+1)^2).
(End)

More terms from Vincenzo Librandi, Sep 19 2015

A051525 Third unsigned column of triangle A051338.

Original entry on oeis.org

0, 0, 1, 21, 335, 5000, 74524, 1139292, 18083484, 299705400, 5198985576, 94461323616, 1797180658272, 35776357096896, 744402741205824, 16169795109262080, 366214212167489280, 8636605663418933760

Offset: 0

Wolfdieter Lang

From Johannes W. Meijer, Oct 20 2009: (Start)
The asymptotic expansion of the higher order exponential integral E(x,m=3,n=6) ~ exp(-x)/x^3*(1 - 21/x + 335/x^2 - 5000/x^3 + 74524/x^4 - 1139292/x^5 + ...) leads to the sequence given above. See A163931 and A163932 for more information.
(End)

Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051338.

Cf. A001725 (m=0), A051524 (m=1) unsigned columns.

a(n) = A051338(n, 2)*(-1)^n; e.g.f.: (log(1-x))^2/(2*(1-x)^6).

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = |f(n,2,6)|, for n>=1. - Milan Janjic, Dec 21 2008

A051546 Third unsigned column of triangle A051339.

Original entry on oeis.org

0, 0, 1, 24, 431, 7155, 117454, 1961470, 33775244, 603682596, 11235811536, 218055250512, 4413843664416, 93156324734304, 2048591287486080, 46898664421553280, 1116592842912341760, 27618683992928743680

Offset: 0

Wolfdieter Lang

From Johannes W. Meijer, Oct 20 2009: (Start)
The asymptotic expansion of the higher order exponential integral E(x,m=3,n=7) ~ exp(-x)/x^3*(1 - 24/x + 431/x^2 - 7155/x^3 + 117454/x^4 + ...) leads to the sequence given above. See A163931 and A163932 for more information.
(End)

Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051339.

Cf. A001730 (m=0), A051545 (m=1) unsigned columns.

a(n) = A051339(n, 2)*(-1)^n; e.g.f.: (log(1-x))^2/(2*(1-x)^7).

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = |f(n,2,7)|, for n>=1. - Milan Janjic, Dec 21 2008

A051561 Third unsigned column of triangle A051379.

Original entry on oeis.org

0, 0, 1, 27, 539, 9850, 176554, 3197348, 59354028, 1137868848, 22614500016, 466814750688, 10015620672672, 223359393479040, 5175622796192640, 124533006364442880, 3109120944743427840, 80473740053567016960

Offset: 0

Wolfdieter Lang

From Johannes W. Meijer, Oct 20 2009: (Start)
The asymptotic expansion of the higher order exponential integral E(x,m=3,n=8) ~ exp(-x)/x^3*(1 - 27/x + 539/x^2 - 9850/x^3 + 176554/x^4 + ...) leads to the sequence given above. See A163931 and A163932 for more information.
(End)

Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051379.

Cf. A049388 (m=0), A051560 (m=1) unsigned columns.

With[{nn=20},CoefficientList[Series[(Log[1-x])^2/(2(1-x)^8),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 10 2013 *)

a(n) = A051379(n, 2)*(-1)^n; e.g.f.: ((log(1-x))^2)/(2*(1-x)^8).

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = |f(n,2,8)|, for n>=1. - Milan Janjic, Dec 21 2008

A074246 Triangle of coefficients, read by rows, where the n-th row forms the polynomial P(n,x) = {Sum_{k=1..n} 1/(k+x)}*{Product_{k=1..n} (k+x)}.

Original entry on oeis.org

1, 3, 2, 11, 12, 3, 50, 70, 30, 4, 274, 450, 255, 60, 5, 1764, 3248, 2205, 700, 105, 6, 13068, 26264, 20307, 7840, 1610, 168, 7, 109584, 236248, 201852, 89796, 22680, 3276, 252, 8, 1026576, 2345400, 2171040, 1077300, 316365, 56700, 6090, 360, 9

Offset: 1

Paul D. Hanna, Sep 19 2002

The n-th row polynomial, P(n,x), has ordered zeros {z_k < z_(k+1), 0

The higher-order exponential integrals E(x,m,n) are defined in A163931 and the asymptotic expansion of E(x,m=2,n) can be found in A028421. We determined with the latter that E(x,m=2,n+1) = (exp(-x)/x^2)*(1 - (3+2*n)/x + (11+12*n+3*n^2)/x^2 - (50+70*n+30*n^2+ 4*n^3)/x^3 + .... ). The polynomial coefficients in the numerators lead to the coefficients of the triangle given above. The numerators of the o.g.f.s of the right hand columns of this triangle lead for z = 1 to A001147. - Johannes W. Meijer, Oct 16 2009

			Polynomials begin:
P(1,x) = 1,
P(2,x) = 3 + 2x,
P(3,x) = 11 + 12x + 3x^2,
P(4,x) = 50 + 70x + 30x^2 + 4x^3,
P(5,x) = 274 + 450x + 255x^2 + 60x^3 + 5x^4,
P(6,x) = 1764 + 3248x + 2205x^2 + 700x^3 + 105x^4 + 6x^5,
P(7,x) = 13068 + 26264x + 20307x^2 + 7840x^3 + 1610x^4 + 168x^5 + 7x^6,
P(8,x) = 109584 + 236248x + 201852x^2 + 89796x^3 + 22680x^4 + 3276x^5 + 252x^6 + 8x^7,
P(9,x) = 1026576 + 2345400x + 2171040x^2 + 1077300x^3 + 316365x^4 + 56700x^5 + 6090x^6 + 360x^7 + 9x^8,
P(10,x) = 10628640 + 25507152x + 25228500x^2 + 13667720x^3 + 4510275x^4 + 946638x^5 + 127050x^6 + 10560x^7 + 495x^8 + 10x^9, ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

See references and formulas at A000254, A001705. Cf. A028421.

A027480 is the second right hand column. - Johannes W. Meijer, Oct 16 2009

with(combinat): A074246 := proc(n,m): (-1)^(n+m)*binomial(m,1)*stirling1(n+1,m+1) end: seq(seq(A074246(n,m),m=1..n),n=1..9); # Johannes W. Meijer, Oct 16 2009, Revised Sep 09 2012

p[n_, x_] := Sum[1/(k+x), {k, 1, n}] Product[k+x, {k, 1, n}] ; Flatten[Table[ CoefficientList[ p[n, x] // Simplify[#, ComplexityFunction -> Length] &, x], {n, 1, 9}]] (* Jean-François Alcover, May 04 2011 *)

P(n) = Vecrev(sum(k=1, n, prod(k=1, n, (k+x))/(k+x)));
for (n=1, 10, print(P(n))) \\ Michel Marcus, Jan 22 2017

First column is A000254 (Stirling numbers of first kind s(n, 2): a(n+1)=(n+1)*a(n)+n!), while sum of rows is A001705 (generalized Stirling numbers). Also related to Harmonic numbers: P(n, 0)=n!*H(n), H(n)=harmonic number.
T(n,k) = (-1)^(n+k)*k*Stirling1(n+1,k+1). - Johannes W. Meijer, Oct 16 2009
E.g.f.: 1/(1 - z)^(x+1)*log(1/(1 - z)). Cf. A028421. - Peter Bala, Jan 06 2015

A367732 Decimal expansion of Sum_{k>=1} (-1)^(k+1) / (k^2 * k!).

Original entry on oeis.org

8, 9, 1, 2, 1, 2, 7, 9, 8, 1, 1, 1, 3, 0, 2, 3, 7, 6, 0, 6, 9, 8, 5, 7, 8, 6, 2, 4, 5, 5, 3, 5, 4, 6, 2, 5, 1, 6, 9, 6, 0, 1, 2, 5, 1, 1, 9, 7, 9, 4, 8, 3, 2, 4, 8, 6, 8, 7, 7, 4, 5, 4, 1, 2, 3, 1, 6, 6, 5, 2, 5, 5, 7, 8, 8, 0, 6, 9, 7, 2, 2, 8, 7, 3, 7, 5, 0, 0, 3, 5, 7, 0, 7, 1, 8, 2, 2, 5, 1, 8

Offset: 0

Ilya Gutkovskiy, Nov 28 2023

			0.89121279811130237606985786245535...

Cf. A002775, A163931, A239069, A355234, A367731.

RealDigits[HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -1], 10, 100][[1]]

A051563 Third unsigned column of triangle A051380.

Original entry on oeis.org

0, 0, 1, 30, 659, 13145, 255424, 4985316, 99236556, 2030997852, 42924478536, 938984014584, 21283428847680, 500043968498880, 12176238355176960, 307176581692097280, 8023946251816984320, 216880826334455750400

Offset: 0

Wolfdieter Lang

From Johannes W. Meijer, Oct 20 2009: (Start)
The asymptotic expansion of the higher order exponential integral E(x,m=3,n=9) ~ exp(-x)/x^3*(1 - 30/x + 659/x^2 - 13145/x^3 + 255424/x^4 + ...) leads to the sequence given above. See A163931 and A163932 for more information.
(End)

Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051380.

Cf. A049389 (m=0), A051562 (m=1) unsigned columns.

a(n) = A051380(n, 2)*(-1)^n; e.g.f.: ((log(1-x))^2)/(2*(1-x)^9).

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = |f(n,2,9)|, for n>=1. - Milan Janjic, Dec 21 2008

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A001722 Generalized Stirling numbers.

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A001723 Generalized Stirling numbers.

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A001724 Generalized Stirling numbers.

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A262343 Numerator of 3*(1-2/n), for n >= 3.

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A051525 Third unsigned column of triangle A051338.

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A051546 Third unsigned column of triangle A051339.

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A051561 Third unsigned column of triangle A051379.

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A074246 Triangle of coefficients, read by rows, where the n-th row forms the polynomial P(n,x) = {Sum_{k=1..n} 1/(k+x)}*{Product_{k=1..n} (k+x)}.

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