A001711 -id:A001711 - cp's OEIS frontend

A001714 Generalized Stirling numbers.

1, 25, 445, 7140, 111769, 1767087, 28699460, 483004280, 8460980836, 154594537812, 2948470152264, 58696064973000, 1219007251826064, 26390216795274288, 594982297852020288, 13955257961738192448, 340154857108405040256, 8606960634143667938688

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher-order exponential integral E(x,m=5,n=3) ~ exp(-x)/x^5*(1 - 25/x + 445/x^2 - 7140/x^3 + 111769/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
From Petros Hadjicostas, Jun 13 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) and Mitrinovic and Mitrinovic (1962) using slightly different notation.
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_0^0(a,b) = 1, R_1^0(a,b) = a, R_1^1(a,b) = 1, and R_n^m(a,b) = 0 for n < m.
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) for n, m >= 0.
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+4}^4(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
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 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].
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.

Cf. A001711, A001712, A001713, A048994, A163931, A163932.

nn = 24; t = Range[0, nn]! CoefficientList[Series[Log[1 - x]^4/(24*(1 - x)^3), {x, 0, nn}], x]; Drop[t, 4] (* T. D. Noe, Aug 09 2012 *)

a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(k+4, 4) * 3^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,4,3)| for n >= 4. - Milan Janjic, Dec 21 2008
From Petros Hadjicostas, Jun 14 2020: (Start)
a(n) = [x^4] Product_{r=0}^{n+3} (x + 3 + r) = (Product_{r=0}^{n+3} (r+3)) * Sum_{0 <= i < j < k < m <= n+3} 1/((3+i)*(3+j)*(3+k)*(3+m)).
E.g.f.: Sum_{n>=0} a(n)*x^(n+4)/(n+4)! = (log(1 - x))^4/(1 - x)^3/24.
Since a(n) = R_{n+4}^4(a=-3, b=-1), A001713(n) = R_{n+3}^3(a=-3,b=-1), A001712(n) = R_{n+2}^2(a=-3, b=-1), and A001711(n) = R_{n+1}^1(a=-3,b=-1), the equation R_{n+4}^4(a=-3,b=-1) = R_{n+3}^3(a=-3,b=-1) + (n+6)*R_{n+3}^4(a=-3,b=-1) implies the following:
(i) a(n) = A001713(n) + (n+6)*a(n-1) for n >= 1.
(ii) a(n) = A001712(n) + (2*n+11)*a(n-1) - (n+5)^2*a(n-2) for n >= 2.
(iii) a(n) = A001711(n) + 3*(n+5)*a(n-1) - (3*n^2+27*n+61)*a(n-2) + (n+4)^3*a(n-3) for n >= 3.
(iv) a(n) = (n+2)!/2 + 2*(2*n+9)*a(n-1) - (6*n^2+48*n+97)*a(n-2) + (2*n+7)*(2*n^2+14*n+25)*a(n-3) - (n+3)^4*a(n-4) for n >= 4.
(v) By taking the difference a(n) - (n+2)*a(n-1), and using (iv) above, we get a 5th-order linear recurrence with polynomial coefficients of degree at most 5. We omit the details. (End)

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

A138771 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} whose 2nd cycle has k entries; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements (n>=1; 0<=k<=n-1). For example, 1432=(1)(24)(3) has 2 entries in the 2nd cycle; 3421=(1324) has 0 entries in the 2nd cycle.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 11, 5, 2, 24, 50, 26, 14, 6, 120, 274, 154, 94, 54, 24, 720, 1764, 1044, 684, 444, 264, 120, 5040, 13068, 8028, 5508, 3828, 2568, 1560, 720, 40320, 109584, 69264, 49104, 35664, 25584, 17520, 10800, 5040

Offset: 1

Emeric Deutsch, Apr 10 2008

T(n,0)=(n-1)!=A000142(n-1).
T(n,1)=A000254(n-1).
T(n,2)=A001705(n-2).
T(n,3)=2*A001711(n-4).
T(n,4)=6*A001716(n-5).
T(n,n-1)=(n-2)! (n>=2).
Sum(kT(n,k),k=0..n-1)=(n-1)!(n-1)(n+2)/4=A138772(n).

			T(4,2)=5 because we have (1)(23)(4), (1)(24)(3), (13)(24), (12)(34) and (14)(23).
Triangle starts;
1;
1,1;
2,3,1;
6,11,5,2;
24,50,26,14,6;
120,274,154,94,54,24;

Cf. A000142, A000254, A001705, A001711, A001716, A138772.
From Johannes W. Meijer, Oct 16 2009: (Start)
A000142 equals for n=>1 the row sums.
a(n) = A165680(n) * A165675(n-1).
(End)

T:=proc (n,k) if k = 0 then factorial(n-1) elif n <= k then 0 else (n-1)*T(n-1, k)+factorial(n-2) end if end proc: for n to 9 do seq(T(n, k), k=0..n-1) end do;

T(n,k)=(n-1)T(n-1,k)+(n-2)! (1<=k<=n-1). The row generating polynomials P[n](t) satisfy: P[n+1](t)=nP[n](t)+(n-1)!(t+t^2+...+t^n).

A196845 Table of elementary symmetric function a_k(3,4,...,n+2) (no 1 and 2).

Original entry on oeis.org

1, 1, 3, 1, 7, 12, 1, 12, 47, 60, 1, 18, 119, 342, 360, 1, 25, 245, 1175, 2754, 2520, 1, 33, 445, 3135, 12154, 24552, 20160, 1, 42, 742, 7140, 40369, 133938, 241128, 181440, 1, 52, 1162, 14560, 111769, 537628, 1580508, 2592720, 1814400, 1, 63, 1734, 27342, 271929, 1767087, 7494416, 19978308, 30334320, 19958400

Offset: 0

Wolfdieter Lang, Oct 26 2011

For the symmetric functions a_k see a comment in A196841.
In general the triangle S_{i,j}(n,k), n>=k>=0, 1<=i=i as a_k(1,2,...,i-1,i+1,...,j-1,j+1,...,n+2).
  a_0():=1. The present triangle is S_{1,2}(n,k) (no 1 and 2 admitted).

			n\k  0   1    2     3     4       5       6       7  ...
0:   1
1:   1   3
2:   1   7   12
3:   1  12   47    60
4:   1  18  119   342   360
5:   1  25  245  1175  2754    2520
6:   1  33  445  3135 12154   24552   20160
7:   1  42  742  7140 40369  133938  241128  181440
...
a(3,2) = a_2(3,4,5) = 3*4+3*5+4*5 = 47.
a(3,2) = 1*(|s(6,4)| - (1*14 + 2*13)) + 2*(|s(6,6)| -(1*0+2*0)) = 85 - 40 + 2(1-0) = 47.
a(4,3) =  a_3(3,4,5,6) = 3*4*5+3*4*6+3*5*6+4*5*6 = 342.
a(4,3) = 1*(|s(7,4)| - (1*155 + 2*137)) + 2*(|s(7,6)| - (1*1 + 2*1)) = 735-429+2*(21-3) = 342.

Cf. A196841, A048994, A145324, A001710 (diagonal), A001711 (1st subdiagonal), A001712 (2nd subdiagonal), A055998 (k=1), A024183 (k=2), A024184 (k=3), A024185 (k=4).

a(n,k) = 0 if n=0, k=0,...,n, with the elementary symmetric function a_k (see the comment above).

a(n,k) = sum(2^k*( |s(n+3,n+3-k+2*p)| -(S_1(n+1,k-1-2*p) +2*S_2(n+1,k-1-2*p))), p=0..floor(k/2)), with the Stirling numbers of the first kind s(n,m) = A048994(n,m), and the number triangles S_1(n,k)= A145324(n+1,k+1) and S_2(n,k) = A196841(n,k).

A346845 E.g.f.: log(1 + x) / (1 - x)^3.

Original entry on oeis.org

1, 5, 29, 186, 1374, 11352, 105048, 1070640, 11978640, 145558080, 1914027840, 27035890560, 408891369600, 6585851059200, 112656894336000, 2038285492992000, 38915729475840000, 781515776369664000, 16475855040820224000, 363685261902133248000, 8391522945839007744000

Offset: 1

Ilya Gutkovskiy, Aug 05 2021

nonn

Cf. A000217, A001711, A024167, A109792, A346846, A346847.

nmax = 21; CoefficientList[Series[Log[1 + x]/(1 - x)^3, {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! Sum[(-1)^(k + 1) Binomial[n - k + 2, 2]/k , {k, 1, n}], {n, 1, 21}]
Table[n!*(((-1)^n*(2*n + 5) - 4*n - 5)/8 + (n+1)*(n+2)*(Log[2] - (-1)^n * LerchPhi[-1, 1, 1 + n])/2), {n, 1, 21}] // Simplify (* Vaclav Kotesovec, Aug 06 2021 *)

my(x='x+O('x^25)); Vec(serlaplace(log(1+x)/(1-x)^3)) \\ Michel Marcus, Aug 06 2021

a(n) = n! * Sum_{k=1..n} (-1)^(k+1) * binomial(n-k+2,2) / k.

a(n) ~ log(2) * n^2 * n! / 2. - Vaclav Kotesovec, Aug 06 2021

A122057 a(n) = (n+1)! * (H(n+1) - H(2)), where H(n) are the harmonic numbers.

Original entry on oeis.org

0, 2, 14, 94, 684, 5508, 49104, 482256, 5185440, 60668640, 767940480, 10462227840, 152698210560, 2377651449600, 39350097561600, 689874448435200, 12773427499929600, 249097496204390400, 5103595024496640000, 109608397522606080000, 2462475687669043200000

Offset: 1

Roger L. Bagula, Sep 14 2006

nonn

Former title (corrected): A Legendre-based recurrence sequence. Let b(n) = ((4*n+2)*x -(2*n+1) )/(n+1)*b(n-1) - (n/(n+1))*b(n-2), where x=1, then a(n) = (n+1)!*b(n)/6. - G. C. Greubel, Oct 03 2019

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964, 9th Printing (1970), pp. 782

G. C. Greubel, Table of n, a(n) for n = 1..445
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A001008, A002805, A001705, A001711.

List([1..30], n-> Factorial(n+1)*(Sum([1..n+1], k-> 1/k) - 3/2) ); # G. C. Greubel, Oct 03 2019

[Factorial(n+1)*(HarmonicNumber(n+1) - 3/2): n in [1..30]]; // G. C. Greubel, Oct 03 2019

a:=n-> (n+1)!*add(1/k,k=3..n+1): seq(a(n),n=1..30); # Gary Detlefs, Jul 15 2010

x=1; b[1]:=0; b[2]:=2; b[n_]:= b[n]= ((-2*n-1) +(4*n+2)*x)/(n+1)*b[n-1] - (n/(n+1))*b[n-2]; Table[b[n]*(n+1)!/6, {n,30}]
Table[(n+1)!*(HarmonicNumber[n+1] - 3/2), {n,30}] (* G. C. Greubel, Oct 03 2019 *)

vector(30, n, (n+1)!*(sum(k=1,n+1, 1/k) - 3/2) ) \\ G. C. Greubel, Oct 03 2019

[factorial(n+1)*(harmonic_number(n+1) - 3/2) for n in (1..30)] # G. C. Greubel, Oct 03 2019

Let b(n) = ((-2*n-1) +(4*n+2)*x)/(n+1)*b(n-1) - (n/(n+1))*b(n-2) with x=1, then a(n) = b(n)*(n+1)!/6.
a(n) = (n+1)! * Sum_{k=3..n+1} 1/k. - Gary Detlefs, Jul 15 2010
a(n) = 2*A001711(n-2) for n >= 2. - Pontus von Brömssen, Jan 04 2025

If all terms are really negative, sequence should probably be negated. - N. J. A. Sloane, Oct 01 2006

Negated terms and edited by G. C. Greubel, Oct 03 2019

A325139 Triangle T(n, k) = [t^n] Gamma(n + k + m + t)/Gamma(k + m + t) for m = 2 and 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 2, 1, 6, 7, 1, 24, 47, 15, 1, 120, 342, 179, 26, 1, 720, 2754, 2070, 485, 40, 1, 5040, 24552, 24574, 8175, 1075, 57, 1, 40320, 241128, 305956, 134449, 24885, 2086, 77, 1, 362880, 2592720, 4028156, 2231012, 541849, 63504, 3682, 100, 1

Offset: 0

Peter Luschny, Apr 15 2019

			0:        1;
1:        2,        1;
2:        6,        7,        1;
3:       24,       47,       15,        1;
4:      120,      342,      179,       26,        1;
5:      720,     2754,     2070,      485,       40,       1;
6:     5040,    24552,    24574,     8175,     1075,      57,      1;
7:    40320,   241128,   305956,   134449,    24885,    2086,     77,    1;
8:   362880,  2592720,  4028156,  2231012,   541849,   63504,   3682,  100,   1;
9:  3628800, 30334320, 56231712, 37972304, 11563650, 1768809, 142632, 6054, 126, 1;
A:  A000142,  A001711,  A001717,  A001723, ...

Row sums are A325140.
Columns are: A000142,  A001711,  A001717,  A001723.
Family: A307419 (m=0), A325137 (m=1), this sequence (m=2).

T := (n, k) -> add(binomial(j+k, k)*(k+2)^j*abs(Stirling1(n, j+k)), j=0..n-k):
seq(seq(T(n, k), k=0..n), n=0..8);
# Note that for n > 16 Maple fails (at least in some versions) to compute the
# terms properly. Inserting 'simplify' or numerical evaluation might help.
A325139Row := proc(n) local ogf, ser; ogf := (n, k) -> GAMMA(n+k+2+x)/GAMMA(k+2+x);
ser := (n, k) -> series(ogf(n,k), x, k+2); seq(coeff(ser(n,k), x, k), k=0..n) end:
seq(A325139Row(n), n=0..9);

T(n, k) = Sum_{j=0..n-k} binomial(j+k, k)*abs(Stirling1(n, j+k))*(k+2)^j.

A376582 Triangle of generalized Stirling numbers.

Original entry on oeis.org

1, 5, 1, 26, 7, 1, 154, 47, 9, 1, 1044, 342, 74, 11, 1, 8028, 2754, 638, 107, 13, 1, 69264, 24552, 5944, 1066, 146, 15, 1, 663696, 241128, 60216, 11274, 1650, 191, 17, 1, 6999840, 2592720, 662640, 127860, 19524, 2414, 242, 19, 1, 80627040, 30334320, 7893840, 1557660, 245004, 31594, 3382, 299, 21, 1

Offset: 0

Igor Victorovich Statsenko, Sep 29 2024

			Triangle starts:
[0]       1;
[1]       5,       1;
[2]      26,       7,       1;
[3]     154,      47,       9,        1;
[4]    1044,     342,      74,       11,       1;
[5]    8028,    2754,     638,      107,      13,     1;
[6]   69264,   24552,    5944,     1066,     146,    15,    1;
[7]  663696,  241128,   60216,    11274,    1650,   191,   17,    1;

Column k: A001705 (k=0), A001711 (k=1), A001716 (k=2), A001721 (k=3), A051524 (k=4), A051545 (k=5), A051560 (k=6).

Cf. A094587 and A173333 for m=0.

T:=(m,n,k)->add(Stirling1(i+m,m)*binomial(n+m+1,n-k-i)*(n+m-k)!/(i+m)!,i=0..n-k): m:=1: seq(seq(T(m,n,k), k=0..n), n=0..10);

T(m,n,k) = Sum_{i=0..n-k} Stirling1(i+m,m)*binomial(n+m+1,n-k-i)*(n+m-k)!/(i+m)!, for m=1.

A024187 n-th elementary symmetric function of 3,4,...,n+3.

Original entry on oeis.org

7, 47, 342, 2754, 24552, 241128, 2592720, 30334320, 383970240, 5231113920, 76349105280, 1188825724800, 19675048780800

Offset: 1

Clark Kimberling

nonn

Conjecture: essentially the same as A001711. - Ralf Stephan, Dec 30 2004

A292717 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. -log(1 - x)/(1 - x)^k.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 5, 11, 6, 0, 1, 7, 26, 50, 24, 0, 1, 9, 47, 154, 274, 120, 0, 1, 11, 74, 342, 1044, 1764, 720, 0, 1, 13, 107, 638, 2754, 8028, 13068, 5040, 0, 1, 15, 146, 1066, 5944, 24552, 69264, 109584, 40320, 0, 1, 17, 191, 1650, 11274, 60216, 241128, 663696, 1026576, 362880

Offset: 0

Ilya Gutkovskiy, Sep 21 2017

			E.g.f. of column k: A_k(x) = x/1! + (2*k + 1)*x^2/2! + (3*k^2 + 6*k + 2)*x^3/3! + 2*(2*k^3 + 9*k^2 + 11*k + 3)*x^4/4! + ...
Square array begins:
   0,    0,     0,     0,     0,      0,  ...
   1,    1,     1,     1,     1,      1,  ...
   1,    3,     5,     7,     9,     11,  ...
   2,   11,    26,    47,    74,    107,  ...
   6,   50,   154,   342,   638,   1066,  ...
  24,  274,  1044,  2754,  5944,  11274,  ...

Columns k=0..11 give A104150, A000254, A001705, A001711 (with offset 1), A001716 (with offset 1),  A001721 (with offset 1), A051524, A051545, A051560, A051562, A051564, A203147.
Rows n=0..3 give  A000004, A000012, A005408, A080663 (with offset 0).
Main diagonal gives A058806.

Table[Function[k, n! SeriesCoefficient[-Log[1 - x]/(1 - x)^k, {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

E.g.f. of column k: -log(1 - x)/(1 - x)^k.

cp's OEIS Frontend

A001714 Generalized Stirling numbers.

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A196845 Table of elementary symmetric function a_k(3,4,...,n+2) (no 1 and 2).

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A346845 E.g.f.: log(1 + x) / (1 - x)^3.

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A122057 a(n) = (n+1)! * (H(n+1) - H(2)), where H(n) are the harmonic numbers.

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A325139 Triangle T(n, k) = [t^n] Gamma(n + k + m + t)/Gamma(k + m + t) for m = 2 and 0 <= k <= n, read by rows.

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A376582 Triangle of generalized Stirling numbers.

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A024187 n-th elementary symmetric function of 3,4,...,n+3.

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A292717 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. -log(1 - x)/(1 - x)^k.