A001701 -id:A001701 - cp's OEIS frontend

A001712 Generalized Stirling numbers.

1, 12, 119, 1175, 12154, 133938, 1580508, 19978308, 270074016, 3894932448, 59760168192, 972751628160, 16752851775360, 304473528961920, 5825460745532160, 117070467915075840, 2465958106403712000, 54336917746726272000, 1250216389189281024000

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=3,n=3) ~ exp(-x)/x^3*(1 - 12/x + 119/x^2 - 1175/x^3 + 12154/x^4 - 133938/x^5 + ...) leads to the sequence given above. See A163931 and A163932 for more information. - Johannes W. Meijer, Oct 20 2009
From Petros Hadjicostas, Jun 11 2020: (Start)
For nonnegative integers n, m and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) using slightly different notation. They were further examined by Mitrinovic and Mitrinovic (1962).
These numbers are defined via the g.f. Product_{r=0..n-1} (x - (a + b*r)) = Sum_{m=0..n} R_n^m(a,b)*x^m for n >= 0.
As a result, R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b) for n >= m >= 1 with R_1^0(a,b) = a, R_1^1(a,b) = 1, and R_n^m(a,b) = 0 for n < m. (Because an empty product is by definition 1, we may let R_0^0(a,b) = 1.)
With a = 0 and b = 1, we get the Stirling numbers of the first kind S1(n,m) = R_n^m(a=0, b=1) = A048994(n,m). (Array A008275 is the same as array A048994 but with no zero row and no zero column.)
We have R_n^m(a,b) = Sum_{k=0}^{n-m} (-1)^k * a^k * b^(n-m-k) * binomial(m+k, k) * S1(n, m+k) for n >= m >= 0.
For the current sequence, a(n) = R_{n+2}^2(a=-3, b=-1) for n >= 0. (End)

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
Matt Davis, Quadrant Marked Mesh Patterns and the r-Stirling Numbers, arXiv preprint arXiv:1412.0345 [math.CO], 2014.
Matt Davis, Quadrant Marked Mesh Patterns and the r-Stirling Numbers, J. Int. Seq. 18 (2015), #15.10.1.
Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, arXiv preprint arXiv:1201.1323 [math.CO], 2012.
Sergey Kitaev and Jeffrey Remmel, Quadrant Marked Mesh Patterns, J. Int. Seq. 15 (2012), #12.4.7.
D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356. [The numbers R_n^m(a,b) are introduced.]
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
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), 1-77 [jstor stable version].
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]

Cf. A001711, A008275, A048994, A163931, A163932.

A001712 := proc(n)
    add((-1)^(n+k)*binomial(k+2, 2)*3^k*Stirling1(n+2, k+2), k=0..n) ;
end proc:
seq(A001712(n), n=0..10) ; # R. J. Mathar, Jun 09 2018

nn = 22; t = Range[0, nn]! CoefficientList[Series[Log[1 - x]^2/(2*(1 - x)^3), {x, 0, nn}], x]; Drop[t, 2] (* T. D. Noe, Aug 09 2012 *)

a(n) = sum(k=0, n, (-1)^(n+k)*binomial(k+2, 2)*3^k*stirling(n+2, k+2, 1)) \\ Michel Marcus, Jan 20 2016

b(n) = prod(r=0, n+1, r+3);
c(n) = sum(i=0, n+1, sum(j=i+1, n+1, 1/((3+i)*(3+j))));
for(n=0, 18, print1(b(n)*c(n),",")) \\ Petros Hadjicostas, Jun 11 2020

a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+2, 2)*3^k*Stirling1(n+2, k+2). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
E.g.f.: (1 - 7*log(1 - x) + 6*log(1 - x)^2)/(1 - x)^5. - Vladeta Jovovic, Mar 01 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,2,3)|, for n >= 2. [Milan Janjic, Dec 21 2008]
Conjecture: a(n) + 3*(-n-3)*a(n-1) + (3*n^2 + 15*n + 19)*a(n-2) - (n+2)^3*a(n-3)=0. - R. J. Mathar, Jun 09 2018
From Petros Hadjicostas, Jun 11 2020: (Start)
a(n) = [x^2] Product_{r=0}^{n+1} (x + 3 + r) = (Product_{r=0}^{n+1} (r+3)) * Sum_{0 <= i < j <= n+1} 1/((3+i)*(3+j)).
Since a(n) = R_{n+2}^2(a=-3, b=-1) and A001711(n) = R_{n+1}^1(a=-3,b=-1), the equation R_{n+2}^2(a=-3,b=-1) = R_{n+1}^1(a=-3,b=-1) + (n+4)*R_{n+1}^2(a=-3,b=-1) implies the following:
(i) a(n) = A001711(n) + (n+4)*a(n-1) for n >= 1.
(ii) a(n) = (n+2)!/2 + (2*n+7)*a(n-1) - (n+3)^2*a(n-2) for n >= 2.
(iii) R. J. Mathar's recurrence above. (End)

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

A001707 Generalized Stirling numbers.

Original entry on oeis.org

1, 14, 155, 1665, 18424, 214676, 2655764, 34967140, 489896616, 7292774280, 115119818736, 1922666722704, 33896996544384, 629429693586048, 12283618766690304, 251426391808144896, 5387217520095244800, 120615281647055884800, 2817014230489985049600

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=4,n=2) ~ exp(-x)/x^4*(1 - 14/x + 155/x^2 - 1665/x^3 + 18424/x^4 - 214676/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 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]

nn = 23; t = Range[0, nn]! CoefficientList[Series[-Log[1 - x]^3/(6*(1 - x)^2), {x, 0, nn}], x]; Drop[t, 3] (* T. D. Noe, Aug 09 2012 *)

a(n) = sum(k=0, n, (-1)^(n+k)*binomial(k+3, 3)*2^k*stirling(n+3, k+3, 1)); \\ Michel Marcus, Jan 01 2023

E.g.f.: - log ( 1 - x )^3 / 6 ( x - 1 )^2.
a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+3, 3)*2^k*Stirling1(n+3, k+3). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 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-3) = |f(n,3,2)|, for n>=3. [From Milan Janjic, Dec 21 2008]

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

A001709 Generalized Stirling numbers.

Original entry on oeis.org

1, 27, 511, 8624, 140889, 2310945, 38759930, 671189310, 12061579816, 225525484184, 4392554369840, 89142436976320, 1884434077831824, 41471340993035856, 949385215397800224, 22587683825903611680, 557978742043520648256, 14297219701868137003200

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=6,n=2) ~ exp(-x)/x^6*(1 - 27/x + 511/x^2 - 8624/x^3 + 140889/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 and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
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]

nn = 25; t = Range[0, nn]! CoefficientList[Series[-Log[1 - x]^5/(120*(1 - x)^2), {x, 0, nn}], x]; Drop[t, 5] (* T. D. Noe, Aug 09 2012 *)

a(n) = sum(k=0, n, (-1)^(n+k)*binomial(k+5, 5)*2^k*stirling(n+5, k+5, 1)); \\ Michel Marcus, Jan 01 2023

a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+5, 5)*2^k*Stirling1(n+5, k+5). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
E.g.f.: (6-120*log(1-x)+465*log(1-x)^2-580*log(1-x)^3+261*log(1-x)^4-36*log(1-x)^5)/(6*(1-x)^7). - 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-5) = |f(n,5,2)|, for n>=5. [From Milan Janjic, Dec 21 2008]

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

A001708 Generalized Stirling numbers.

Original entry on oeis.org

1, 20, 295, 4025, 54649, 761166, 11028590, 167310220, 2664929476, 44601786944, 784146622896, 14469012689040, 279870212258064, 5667093514231200, 119958395537083104, 2650594302549806976, 61049697873641191296, 1463708634867162093312, 36482312832434713195776

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=5,n=2) ~ exp(-x)/x^5*(1 - 20/x + 295/x^2 - 4025/x^3 + 54649/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 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), 1-77.
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]

With[{nn=20},Drop[CoefficientList[Series[Log[1-x]^4/(24(1-x)^2),{x,0,nn}], x]Range[0,nn]!,4]] (* Harvey P. Dale, Oct 24 2011 *)

my(x='x+O('x^25)); Vec(serlaplace((log(1-x))^4/(24*(1-x)^2))) \\ Michel Marcus, Feb 04 2022

E.g.f.: ( log ( 1 - x ))^4 / 24 ( 1 - x )^2.
a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+4, 4)*2^k*Stirling1(n+4, k+4). - 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-4) = |f(n,4,2)| for n >= 4. - Milan Janjic, Dec 21 2008

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

A001702 Generalized Stirling numbers.

Original entry on oeis.org

1, 24, 154, 580, 1665, 4025, 8624, 16884, 30810, 53130, 87450, 138424, 211939, 315315, 457520, 649400, 903924, 1236444, 1664970, 2210460, 2897125, 3752749, 4809024, 6101900, 7671950, 9564750, 11831274, 14528304, 17718855, 21472615, 25866400, 30984624

Offset: 1

N. J. A. Sloane

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
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), 1-77.
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), 1-77 [jstor stable version].
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 (7,-21,35,-35,21,-7,1).

For n > 1, a(n) = A145324(n+2,4).

Concatenation([1],List([2..35],n->(n-1)*n*(n+1)*(n+4)*(n^2+7*n+14)/48)); # Muniru A Asiru, Sep 29 2018

[1] cat [n*(n^2-1)*(n+4)*(n^2+7*n+14)/48: n in [2..35]]; // Vincenzo Librandi, Sep 30 2018

A001702 := proc(n)
    if n = 1 then
        1 ;
    else
        (n-1)*n*(n+1)*(n+4)*(n^2+7*n+14)/48 ;
    end if;
end proc: # R. J. Mathar, Sep 23 2016

Join[{1}, Table[(n-1) n (n+1) (n+4) (n^2 + 7 n + 14)/48, {n, 2, 100}]] (* T. D. Noe, Aug 09 2012 *)
CoefficientList[Series[1 +x*(x-4)*(x^2-2*x+6)/(x-1)^7, {x, 0, 100}], x] (* Stefano Spezia, Sep 30 2018 *)
Join[{1},Table[Coefficient[Product[x + j, {j, 2, k}], x, k - 4], {k, 4, 40}]]  (* or *)  Join[{1}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {24, 154, 580, 1665, 4025, 8624, 16884}, 40]] (* Robert A. Russell, Oct 04 2018 *)

vector(50, n, if(n==1, 1, (1/48)*(n-1)*n* (n+1)* (n+4)*(n^2 +7*n +14))) \\G. C. Greubel, Oct 06 2018

a(n) = (1/48)*(n-1)*n*(n+1)*(n+4)*(n^2+7n+14), n > 1.
G.f.: x + x^2*(x-4)*(x^2-2*x+6)/(x-1)^7. - 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-1) = -f(n,n-3,2), for n >= 3. - Milan Janjic, Dec 20 2008
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Colin Barker, Jul 08 2020

A110952 Triangle read by rows: T(n,k) = number of permutations of [n] where the first increasing run has length k and the last increasing run has length n-k-1, 0

Original entry on oeis.org

1, 3, 3, 6, 11, 6, 10, 26, 26, 10, 15, 50, 71, 50, 15, 21, 85, 155, 155, 85, 21, 28, 133, 295, 379, 295, 133, 28, 36, 196, 511, 799, 799, 511, 196, 36, 45, 276, 826, 1519, 1849, 1519, 826, 276, 45, 55, 375, 1266, 2674, 3829, 3829, 2674, 1266, 375, 55, 66, 495, 1860

Offset: 3

David Scambler, Nov 22 2006

Permutations of [n] with exactly 2 descents and the descents are adjacent. Adjusting for initial index: row sums are A045618; first diagonal is A000217, the triangular numbers; 2nd diagonal is A051925; and 3rd diagonal is A001701, generalized Stirling numbers.

			Triangle (beginning with n=3, k=1) is:
   1
   3  3
   6 11  6
  10 26 26 10
  15 50 71 50 15
  ...
For n=5, k = 2: T(5,2) = 11 = permutations of [5] with first run 2 long and last run 5-2-1 = 2 long, namely {14325, 15324, 15423, 24315, 25314, 25413, 34215, 35214, 35412, 45213, 45312}.

Joseph Adams, Table of n, a(n) for n = 3..4755

Cf. A045618, A000217, A051925, A001701, A112858.

T(n,k) = k*C(n,k+1) - C(n,k) + 1.

A024177 a(n) = floor ( (2nd elementary symmetric function of 2,3,...,n+2)/(2+3+...+n+2) ).

Original entry on oeis.org

1, 2, 5, 7, 10, 14, 18, 23, 28, 34, 40, 47, 54, 61, 70, 78, 87, 97, 107, 118, 129, 141, 153, 166, 179, 192, 207, 221, 236, 252, 268, 285, 302, 320, 338, 357, 376, 395, 416, 436, 457, 479, 501, 524

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Table[Floor[1/12 n (3 n^2 + 23 n + 46)/(n + 4)], {n, 44}] (* Ivan Neretin, May 21 2018 *)

a(n) = floor( A001701(n+1)/A000096(n+1) ). - R. J. Mathar, Oct 31 2011
G.f. ( -1-2*x^2+2*x^3-x^4-2*x^6+x^5-x^8+x^9 ) / ( (x^2+1)*(1+x+x^2)*(x^4-x^2+1)*(x-1)^3 ). - R. J. Mathar, Oct 31 2011
a(n) = floor(1/12 n (3 n^2 + 23 n + 46)/(n + 4)). - Ivan Neretin, May 21 2018

A024180 a(n) = floor((3rd elementary symmetric function of 2,3,...,n+3) / (2nd elementary symmetric function of 2,3,...,n+3) ).

Original entry on oeis.org

0, 2, 3, 5, 7, 10, 13, 16, 20, 24, 28, 32, 37, 42, 48, 54, 60, 67, 74, 81, 88, 96, 104, 113, 122, 131, 141, 151, 161, 171, 182, 193, 205, 217, 229, 242, 255, 268, 281, 295, 309, 324, 339, 354, 370, 386, 402, 418, 435, 452, 470, 488

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

s[n_] := 1 + Range[n + 2]
Table[Floor[SymmetricPolynomial[3, s[n]]/SymmetricPolynomial[2, s[n]]], {n, 1,
  46}] (* Clark Kimberling, Sep 23 2016 *)

Empirical g.f.: -x^2*(x^10-2*x^9+x^7+x^4+x^2-x+2) / ((x-1)^3*(x^2+x+1)*(x^6+x^3+1)). - Colin Barker, Aug 16 2014
a(n) = floor(A001702(n+1)/A001701(n+2)). - R. J. Mathar, Sep 23 2016
a(n) = floor((1/2)*n*(5+n)*(n^2 + 9*n + 22)/(3*n^2 + 29*n + 72)). - Ivan Neretin, May 21 2018

Definition corrected by R. J. Mathar, Sep 23 2016

cp's OEIS Frontend

A001712 Generalized Stirling numbers.

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A001707 Generalized Stirling numbers.

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A001709 Generalized Stirling numbers.

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A001708 Generalized Stirling numbers.

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A001702 Generalized Stirling numbers.

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A110952 Triangle read by rows: T(n,k) = number of permutations of [n] where the first increasing run has length k and the last increasing run has length n-k-1, 0

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A024177 a(n) = floor ( (2nd elementary symmetric function of 2,3,...,n+2)/(2+3+...+n+2) ).

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