A048594 -id:A048594 - cp's OEIS frontend

A141412 Triangle c(n,k) of the denominators of coefficients [x^k] P(n,x) of the polynomials P(n,x) of A129891.

1, 2, 1, 3, 1, 1, 4, 12, 2, 1, 5, 6, 4, 1, 1, 6, 180, 8, 6, 2, 1, 7, 10, 15, 2, 6, 1, 1, 8, 560, 240, 240, 6, 4, 2, 1, 9, 1260, 15120, 20, 144, 1, 12, 1, 1, 10, 12600, 672, 945, 32, 240, 8, 3, 2, 1, 11, 1260, 8400, 1512, 3024, 48, 240, 3, 1, 1, 1, 12, 166320, 100800, 64800, 12096, 12096, 480, 360, 4, 12, 2, 1

Offset: 0

Paul Curtz, Aug 04 2008

Polynomials are characteristic polynomials of a particular John Couch Adams matrix.
General term: ( (-1)^(n-j)*C(j, n)*n! ) * Integral_{0..i} (u*(u-1)*(u-2)* ... *(u-n))/(u-j)) du, with 1 <= i,j <= n (see Flajolet et al.).
Denominators are 1, 2, 12, 24, 720 = A091137.
These polynomials come from the explicit case. The less interesting implicit case has the same denominators (see P. Curtz reference).

			Triangle begins:
  1;
  2,   1;
  3,   1,  1;
  4,  12,  2,  1;
  5,   6,  4,  1,  1;
  6, 180,  8,  6,  2,  1;
  7,  10, 15,  2,  6,  1,  1;
  ...

Paul Curtz, Intégration .. note 12, C.C.S.A., Arcueil 1969, p. 61; ibid. pp. 62-65.
P. Flajolet, X. Gourdon, and B. Salvy, Sur une famille de polynômes issus de l'analyse numérique, Gazette des Mathématiciens, 1993, 55, pp. 67-78.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Bakir Farhi, On the derivatives of the integer-valued polynomials, arXiv:1810.07560 [math.NT], 2018.

Cf. A000254, A048594, A129891, A140749 (numerators).

[Denominator(Factorial(k)*StirlingFirst(n, k)/Factorial(n)): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 24 2023

P := proc(n,x) option remember ; if n =0 then 1; else (-1)^n/(n+1)+x*add( (-1)^i/(i+1)*procname(n-1-i,x),i=0..n-1) ; expand(%) ; fi; end:
A141412 := proc(n,k) p := P(n,x) ; denom(coeftayl(p,x=0,k)) ; end: seq(seq(A141412(n,k),k=0..n),n=0..13) ; # R. J. Mathar, Aug 24 2009

p[0]=1; p[n_]:= p[n]= (-1)^n/(n+1) +x*Sum[(-1)^k*p[n-1-k]/(k+1), {k, 0, n-1}];
Denominator[Flatten[Table[CoefficientList[p[n], x], {n,0,11}]]][[1 ;; 72]] (* Jean-François Alcover, Jun 17 2011 *)
Table[Denominator[(k+1)!*StirlingS1[n+1,k+1]/(n+1)!], {n,0,12}, {k,0, n}]//Flatten (* G. C. Greubel, Oct 24 2023 *)

def A141412(n,k): return denominator(factorial(k+1)* stirling_number1(n+1,k+1)/factorial(n+1))
flatten([[A141412(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 24 2023

Conjecture: T(n, k) = d(n+1, k+1), with d(n,k) = denominator(A000254(n, k)*k!/n!) where A000254 are the unsigned Stirling numbers of the 1st kind. See d(n,k) in Farhi link. - Michel Marcus, Oct 18 2018

Equals denominators of A048594(n+1, k+1)/(n+1)!. - G. C. Greubel, Oct 24 2023

Partially edited by R. J. Mathar, Aug 24 2009

A140749 Triangle c(n,k) of the numerators of coefficients [x^k] P(n,x) of the polynomials P(n,x) of A129891.

Original entry on oeis.org

1, -1, 1, 1, -1, 1, -1, 11, -3, 1, 1, -5, 7, -2, 1, -1, 137, -15, 17, -5, 1, 1, -7, 29, -7, 25, -3, 1, -1, 363, -469, 967, -35, 23, -7, 1, 1, -761, 29531, -89, 1069, -9, 91, -4, 1, -1, 7129, -1303, 4523, -285, 3013, -105, 29, -9, 1, 1, -671, 16103, -7645, 31063, -781, 4781, -55, 12, -5, 1

Offset: 0

Paul Curtz, Jul 13 2008

The polynomials P(n,x) are defined in A129891: P(0,x)=1 and

P(n,x) = (-1)^n/(n+1) + x* Sum_{i=0..n-1} (-1)^i*P(n-1-i,x)/(i+1) = Sum_{k=0..n} c(n,k)*x^k.

			The polynomials, for n =0,1,2, ..., are
  P(0, x) = 1;
  P(1, x) = -1/2 + x;
  P(2, x) = 1/3 - x + x^2;
  P(3, x) = -1/4 + 11/12*x - 3/2*x^2 + x^3;
  P(4, x) = 1/5 - 5/6*x + 7/4*x^2 - 2*x^3 + x^4;
  P(5, x) = -1/6 + 137/180*x - 15/8*x^2 + 17/6*x^3 - 5/2*x^4 + x^5;
and the coefficients are
   1;
  -1/2,   1;
   1/3,  -1,       1;
  -1/4,  11/12,   -3/2,   1;
   1/5,  -5/6,     7/4,  -2,     1;
  -1/6, 137/180, -15/8,  17/6,  -5/2,  1;
   1/7,  -7/10,   29/15, -7/2,  25/6, -3,   1;.

Paul Curtz, Gazette des Mathématiciens, 1992, 52, p. 44.
Paul Curtz, Intégration Numérique .. Note 12 du Centre de Calcul Scientifique de l'Armement, Arcueil, 1969. Now in 35170, Bruz.
P. Flajolet, X. Gourdon, and B. Salvy, Sur une famille de polynômes issus de l'analyse numérique, Gazette des Mathématiciens, 1993, 55, pp. 67-78.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Jean-François Alcover, Plot showing roots of P(200,x) in shape of a cardioid

Cf. A048594, A129891, A141412 (denominators).

[Numerator(Factorial(k+1)*StirlingFirst(n+1,k+1)/Factorial(n+1) ): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 24 2023

P := proc(n,x) option remember ; if n =0 then 1; else (-1)^n/(n+1)+x*add( (-1)^i/(i+1)*procname(n-1-i,x),i=0..n-1) ; expand(%) ; fi; end:
A140749 := proc(n,k) p := P(n,x) ; numer(coeftayl(p,x=0,k)) ; end: seq(seq(A140749(n, k),k=0..n),n=0..13) ; # R. J. Mathar, Aug 24 2009

p[0] = 1; p[n_] := p[n] = (-1)^n/(n+1) + x*Sum[(-1)^k*p[n-1-k] / (k+1), {k, 0, n-1}];
Numerator[ Flatten[ Table[ CoefficientList[p[n], x], {n, 0, 11}]]][[1 ;; 69]] (* Jean-François Alcover, Jun 17 2011 *)
Table[Numerator[(k+1)!*StirlingS1[n+1,k+1]/(n+1)!], {n,0,12}, {k,0,n} ]//Flatten (* G. C. Greubel, Oct 24 2023 *)

def A048594(n,k): return (-1)^(n-k)*numerator(factorial(k+1)* stirling_number1(n+1,k+1)/factorial(n+1))
flatten([[A048594(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 24 2023

(n+1)*c(n,k) = (n+1-k)*c(n-1,k) - n*c(n-1, k-1). [Edgard Bavencoffe in 1992]

Equals Numerators of A048594(n+1,k+1)/(n+1)!. - Paul Curtz, Jul 17 2008

Edited and extended by R. J. Mathar, Aug 24 2009

A320079 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k*log(1 - x)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 14, 0, 1, 4, 21, 76, 88, 0, 1, 5, 36, 222, 772, 694, 0, 1, 6, 55, 488, 3132, 9808, 6578, 0, 1, 7, 78, 910, 8824, 55242, 149552, 72792, 0, 1, 8, 105, 1524, 20080, 199456, 1169262, 2660544, 920904, 0, 1, 9, 136, 2366, 39708, 553870, 5410208, 28873800, 54093696, 13109088, 0

Offset: 0

Ilya Gutkovskiy, Oct 05 2018

			E.g.f. of column k: A_k(x) = 1 + k*x/1! + k*(2*k + 1)*x^2/2! + 2*k*(3*k^2 + 3*k + 1)*x^3/3! + 2*k*(12*k^3 + 18*k^2 + 11*k + 3)*x^4/4! + ...
Square array begins:
  1,    1,     1,      1,       1,       1,  ...
  0,    1,     2,      3,       4,       5,  ...
  0,    3,    10,     21,      36,      55,  ...
  0,   14,    76,    222,     488,     910,  ...
  0,   88,   772,   3132,    8824,   20080,  ...
  0,  694,  9808,  55242,  199456,  553870,  ...

Columns k=0..5 give A000007, A007840, A088500, A354263, A354264, A365588.
Main diagonal gives A317171.
Cf. A048594, A048994, A094416, A320080.

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

E.g.f. of column k: 1/(1 + k*log(1 - x)).
A(n,k) = Sum_{j=0..n} |Stirling1(n,j)|*j!*k^j.
A(0,k) = 1; A(n,k) = k * Sum_{j=1..n} (j-1)! * binomial(n,j) * A(n-j,k). - Seiichi Manyama, May 22 2022

A238685 a(n) = n! * A129505(n) * (-1)^(n+1).

Original entry on oeis.org

1, -6, 210, -17640, 2693880, -649479600, 226750764240, -108116216208000, 67478689070432640, -53382381970299782400, 52192613508738839136000, -61794396463636399635072000, 87121906773549083421777792000, -144222462676882552982237906688000

Offset: 1

Reinhard Zumkeller, Mar 02 2014

sign

Reinhard Zumkeller, Table of n, a(n) for n = 1..200

Cf. A000142, A008275, A048594, A129505.

a238685 n = a000142 n * a008275 (2 * n - 1) n

Array[#!*StirlingS1[2 # - 1, #] &, 14] (* Michael De Vlieger, Jan 24 2022 *)

a(n) = n!*stirling(2*n-1,n, 1); \\ Michel Marcus, Jan 24 2022

from math import factorial
from sympy.functions.combinatorial.numbers import stirling
def A238685(n): return factorial(n)*stirling((n<<1)-1,n,kind=1,signed=True) # Chai Wah Wu, Jun 09 2025

a(n) = A000142(n) * A008275(2*n-1,n).

a(n) = A048594(2*n-1,n).

A075183 One half of third column of triangle A075181.

Original entry on oeis.org

1, 11, 105, 1020, 10500, 115920, 1375920, 17539200, 239500800, 3492720000, 54226972800, 893577484800, 15583119552000, 286816578048000, 5557616064000000, 113108602134528000, 2412627824775168000

Offset: 0

Wolfdieter Lang, Sep 19 2002

Also one half of third diagonal of triangle A048594.

Cf. A008275, A048594, A075181.

a(n) = A075181(n+3, 2)/2 = A048594(n+3, n+1)/2, n>=0.

a(n) = (n+1)!*S1(n+3, n+1)/2 with S1(n, m) := A008275(n, m) (Stirling1).

A140219 Denominator of the coefficient [x^1] of the Bernoulli twin number polynomial C(n,x).

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 6, 6, 10, 10, 6, 6, 210, 210, 2, 2, 30, 30, 42, 42, 110, 110, 6, 6, 546, 546, 2, 2, 30, 30, 462, 462, 170, 170, 6, 6, 51870, 51870, 2, 2, 330, 330, 42, 42, 46, 46, 6, 6, 6630, 6630, 22, 22, 30, 30, 798, 798, 290

Offset: 0

Paul Curtz, Jun 23 2008

See A140351 for the main part of the documentation.

Cf. A002427, A006955, A048594, A140351 (numerators).

C := proc(n, x) if n = 0 then 1; else add(binomial(n-1, j-1)* bernoulli(j, x), j=1..n) ; expand(%) ; end if ; end proc:
A140219 := proc(n) coeff(C(n, x), x, 1) ; denom(%) ; end proc:
seq(A140219(n), n=1..80) ; # R. J. Mathar, Sep 22 2011

Table[Sum[Binomial[n, k]*(k+1)*BernoulliB[k], {k, 0, n}], {n, 0, 60}] // Denominator (* Vaclav Kotesovec, Oct 05 2016 *)

makelist(denom(sum((binomial(n, i)*(i+1)*bern(i)), i, 0, n)), n, 0, 20); /* Vladimir Kruchinin, Oct 05 2016 */

a(n) = denominator(sum(i=0, n, binomial(n,i)*(i+1)*bernfrac(i))); \\ Michel Marcus, Oct 05 2016

a(n) = denominator(Sum_{i=0..n} binomial(n,i)*(i+1)*bern(i)). - Vladimir Kruchinin, Oct 05 2016

a(n) = A006955(floor(n/2)). - Georg Fischer, Nov 29 2022

A225475 Triangle read by rows, k!*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 3, 4, 2, 15, 23, 18, 6, 105, 176, 172, 96, 24, 945, 1689, 1900, 1380, 600, 120, 10395, 19524, 24278, 20880, 12120, 4320, 720, 135135, 264207, 354662, 344274, 241080, 116760, 35280, 5040, 2027025, 4098240, 5848344, 6228096, 4993296, 2956800, 1229760

Offset: 0

Peter Luschny, May 19 2013

The Stirling-Frobenius cycle numbers are defined in A225470.

			[n\k][ 0,    1,    2,    3,   4,   5]
[0]    1,
[1]    1,    1,
[2]    3,    4,    2,
[3]   15,   23,   18,    6,
[4]  105,  176,  172,   96,  24,
[5]  945, 1689, 1900, 1380, 600, 120.

Vincenzo Librandi, Rows n = 0..50, flattened
Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

Cf. A028338, A225479 (m=1), A048594.

SFCO[n_, k_, m_] := SFCO[n, k, m] = If[ k > n || k < 0, Return[0], If[ n == 0 && k == 0, Return[1], Return[ k*SFCO[n - 1, k - 1, m] + (m*n - 1)*SFCO[n - 1, k, m]]]]; Table[ SFCO[n, k, 2], {n, 0, 8}, {k, 0, n}] // Flatten  (* Jean-François Alcover, Jul 02 2013, translated from Sage *)

@CachedFunction
def SF_CO(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return k*SF_CO(n-1, k-1, m) + (m*n-1)*SF_CO(n-1, k, m)
for n in (0..8): [SF_CO(n, k, 2) for k in (0..n)]

For a recurrence see the Sage program.
T(n, 0) ~ A001147; T(n, 1) ~ A004041.
T(n, n) ~ A000142; T(n, n-1) ~ A001563.
T(n,k) = A028338(n,k)*A000142(k). - Philippe Deléham, Jun 24 2015

A322470 Expansion of e.g.f. 1/(1 + log(1 + x)/(1 + log(1 + x)^2/(1 + log(1 + x)^3/(1 + ...)))), a continued fraction.

Original entry on oeis.org

1, -1, 3, -8, 4, 236, -3892, 54552, -739440, 9704088, -116868648, 1033709040, 4025264736, -592337009328, 23033374965456, -708140910086400, 19418661884145024, -485092601562305664, 10704418782304457088, -180835985547961196544, 431827528992523301376, 162896031123325288266240

Offset: 0

Ilya Gutkovskiy, Dec 19 2018

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..416
Vaclav Kotesovec, Plot of (abs(a(n))/n!)^(1/n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A007325, A048594, A301923, A322342.

nmax = 21; CoefficientList[Series[1/(1 + ContinuedFractionK[Log[1 + x]^k, 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = Sum_{k=0..n} Stirling1(n,k)*A007325(k)*k!.

A075184 One half of fourth column of triangle A075181.

Original entry on oeis.org

3, 50, 675, 8820, 117600, 1632960, 23814000, 365904000, 5927644800, 101189088000, 1818030614400, 34326452160000, 679990671360000, 14108934647808000, 306113492805120000, 6933770723303424000

Offset: 0

Wolfdieter Lang, Sep 19 2002

Also one half of fourth diagonal of unsigned triangle A048594.

Cf. A008275, A048594, A075181.

a(n) = A075181(n+4, 3)/2 = |A048594(n+4, n+1)|/2, n>=0.

a(n) = -(n+1)!*S1(n+4, n+1)/2 with S1(n, m) := A008275(n, m) (Stirling1).

A075185 One-fourth of fifth column of triangle A075181.

Original entry on oeis.org

6, 137, 2436, 40614, 673470, 11389140, 198793980, 3602823840, 67991283360, 1337641905600, 27440275262400, 586731694348800, 13067437397414400, 302870068070169600, 7298072456298624000

Offset: 0

Wolfdieter Lang, Sep 19 2002

Also one-fourth of fifth diagonal of triangle A048594.

Cf. A008275, A048594, A075181.

a(n) = A075181(n+5, 4)/4 = A048594(n+5, n+1)/4, n>=0.

a(n) = (n+1)!*S1(n+5, n+1)/4 with S1(n, m) := A008275(n, m) (Stirling1).

cp's OEIS Frontend

A141412 Triangle c(n,k) of the denominators of coefficients [x^k] P(n,x) of the polynomials P(n,x) of A129891.

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A140749 Triangle c(n,k) of the numerators of coefficients [x^k] P(n,x) of the polynomials P(n,x) of A129891.

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A320079 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k*log(1 - x)).

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A238685 a(n) = n! * A129505(n) * (-1)^(n+1).

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A075183 One half of third column of triangle A075181.

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A140219 Denominator of the coefficient [x^1] of the Bernoulli twin number polynomial C(n,x).

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A225475 Triangle read by rows, k!*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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