cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 18 results. Next

A240776 Define a square array B(m,n) (m>=0, n>=0) by B(n, n) = A212196(n)/A181131(n), B(n, n+1) = -A212196(n)/A181131(n), B(m, n) = B(m, n-1) + B(m+1, n-1); a(n) = numerator of B(0,n).

Original entry on oeis.org

1, -1, -4, -1, -8, -1, -8, -3, -8, 1, 104, -41, -920, 1767, 20168, -8317, -2022392, 869807, 291391192, -129169263, -2759924456, 250158593, 146772324808, -67632514765, -10164962436952
Offset: 0

Views

Author

Paul Curtz, Apr 12 2014

Keywords

Comments

The array B(m,n) begins:
1, -1, -4/3, -1, -8/15, -1/5, -8/105,...
-2, -1/3, 1/3, 7/15, 1/3, 13/105,...
5/3, 2/3, 2/15, -2/15, -22/105,...
-1, -8/15, -4/15, -8/105,...
7/15, 4/15, 4/21,...
-1/5, -8/105,...
13/105,...
etc.
B(0, n) = 1, -1, -4/3, -1, -8/15, -1/5, -8/105, -3/35, -8/105, 1/35, 104/1155, ... = a(n)/b(n).
The main diagonal is A212196(n)/A181131(n).
The first upper diagonal is -A212196(n)/A181131(n).

Programs

  • Mathematica
    max = 12; t[0] = Table[BernoulliB[n], {n, 0, 2*max}]; t[n_] := t[n] = Differences[t[0], n]; B1[1, 1] = -1/3; B1[n_, n_] := t[n][[n+1]]; B1[m_, n_] /; n == m+1 := B1[m, n] = -B1[m, m]; B1[m_?NonNegative, n_?NonNegative] := B1[m, n] = B1[m, n-1] + B1[m+1, n-1]; B1[, ] = 0; Table[B1[0, n] // Numerator, {n, 0, 2*max}] (* Jean-François Alcover, Apr 14 2014 *)

Extensions

More terms from Jean-François Alcover, Apr 14 2014
Edited by N. J. A. Sloane, May 21 2014

A181131 Denominator of Integral_{x=0..+oo} Polylog(-n, -x)^2 for n > 0, with a(0) = 1.

Original entry on oeis.org

1, 3, 15, 105, 105, 231, 15015, 2145, 36465, 969969, 4849845, 10140585, 10140585, 22287, 3231615, 7713865005, 7713865005, 90751353, 218257003965, 1641030105, 67282234305, 368217318651, 1841086593255, 3762220429695, 63957747304815, 1546231253523
Offset: 0

Views

Author

Vladimir Reshetnikov, Jan 23 2011

Keywords

Comments

These are the denominators of the Bernoulli median numbers (see A212196). - Peter Luschny, May 04 2012

Links

Crossrefs

Cf. A181130, A212196.

Programs

  • Maple
    seq(denom(add(binomial(n,k)*bernoulli(n+k),k=0..n)), n=0..100); # Robert Israel, Jun 02 2015
  • Mathematica
    Table[Denominator[Integrate[PolyLog[-n, -x]^2, {x, 0, Infinity}]], {n, 1, 18}]
max = 25; t[0] = Table[BernoulliB[n], {n, 0, 2*max}]; t[n_] := Differences[t[0], n]; a[n_] := t[n][[n + 1]] // Denominator; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Jul 25 2013, after Peter Luschny *)
  • PARI
    a(n)=denominator(-subst(intformal(polylog(-n,-x)^2),'x,0)) \\ Charles R Greathouse IV, Jul 21 2014
  • Sage
    # uses[BernoulliMedian_list from A212196]
def A181131_list(n):
    return [denominator(q) for q in BernoulliMedian_list(n)]
# Peter Luschny, May 04 2012

Formula

a(n) = denominator((-1)^n/Pi^(2*n)*integral((log(t/(1-t))*log(1-1/t))^n dt,t=0,1)). - [Gerry Martens, May 25 2011]
a(n) = Denominator(Sum_{k=0..n} C(n,k)*Bern(n+k)). - Vladimir Kruchinin, Apr 06 2015

Extensions

Offset set to 0, a(0) and a(19)..a(25) added by Peter Luschny, May 04 2012

A085738 Denominators in triangle formed from Bernoulli numbers.

Original entry on oeis.org

1, 2, 2, 6, 3, 6, 1, 6, 6, 1, 30, 30, 15, 30, 30, 1, 30, 15, 15, 30, 1, 42, 42, 105, 105, 105, 42, 42, 1, 42, 21, 105, 105, 21, 42, 1, 30, 30, 105, 105, 105, 105, 105, 30, 30, 1, 30, 15, 105, 105, 105, 105, 15, 30, 1, 66, 66, 165, 165, 1155, 231, 1155, 165, 165, 66, 66
Offset: 0

Views

Author

N. J. A. Sloane following a suggestion of J. H. Conway, Jul 23 2003

Keywords

Comments

Triangle is determined by rules 0) the top number is 1; 1) each number is the sum of the two below it; 2) it is left-right symmetric; 3) the numbers in each of the border rows, after the first 3, are alternately 0.
Up to signs this is the difference table of the Bernoulli numbers (see A212196). The Sage script below is based on L. Seidel's algorithm and does not make use of a library function for the Bernoulli numbers; in fact it generates the Bernoulli numbers on the fly. - Peter Luschny, May 04 2012

Examples

			Triangle begins
    1
   1/2,   1/2
   1/6,   1/3,   1/6
    0,    1/6,   1/6,     0
  -1/30,  1/30,  2/15,   1/30,  -1/30
    0,   -1/30,  1/15,   1/15,  -1/30,     0
   1/42, -1/42, -1/105,  8/105, -1/105,  -1/42,   1/42
    0,    1/42, -1/21,   4/105,  4/105,  -1/21,   1/42,   0
  -1/30,  1/30, -1/105, -4/105,  8/105,  -4/105, -1/105, 1/30, -1/30

Links

Crossrefs

See A051714/A051715 for another triangle that generates the Bernoulli numbers.
Cf. A085737, A212196.

Programs

  • Mathematica
    t[n_, 0] := (-1)^n BernoulliB[n];
t[n_, k_] := t[n, k] = t[n-1, k-1] - t[n, k-1];
Table[t[n, k] // Denominator, {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 04 2019 *)
  • Sage
    # uses[BernoulliDifferenceTable from A085737]
def A085738_list(n): return [q.denominator() for q in BernoulliDifferenceTable(n)]
A085738_list(6)
# Peter Luschny, May 04 2012

Formula

T(n, 0) = (-1)^n*Bernoulli(n); T(n, k) = T(n-1, k-1) - T(n, k-1) for k=1..n. [Corrected (sign flipped) by R. J. Mathar, Jun 02 2010]
Let U(m, n) = (-1)^(m + n)*T(m+n, n). Then the e.g.f. for U(m, n) is (x - y)/(e^x - e^y). - Ira M. Gessel, Jun 12 2021

A085737 Numerators in triangle formed from Bernoulli numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 1, 1, 0, -1, 1, 2, 1, -1, 0, -1, 1, 1, -1, 0, 1, -1, -1, 8, -1, -1, 1, 0, 1, -1, 4, 4, -1, 1, 0, -1, 1, -1, -4, 8, -4, -1, 1, -1, 0, -1, 1, -8, 4, 4, -8, 1, -1, 0, 5, -5, 7, 4, -116, 32, -116, 4, 7, -5, 5, 0, 5, -5, 32, -28, 16, 16, -28, 32, -5, 5, 0
Offset: 0

Views

Author

N. J. A. Sloane, following a suggestion of J. H. Conway, Jul 23 2003

Keywords

Comments

Triangle is determined by rules 0) the top number is 1; 1) each number is the sum of the two below it; 2) it is left-right symmetric; 3) the numbers in each of the border rows, after the first 3, are alternately 0.
Up to signs this is the difference table of the Bernoulli numbers (see A212196). The Sage script below is based on L. Seidel's algorithm and does not make use of a library function for the Bernoulli numbers; in fact it generates the Bernoulli numbers on the fly. - Peter Luschny, May 04 2012

Examples

			Triangle of fractions begins
    1;
   1/2,   1/2;
   1/6,   1/3,   1/6;
    0,    1/6,   1/6,     0;
  -1/30,  1/30,  2/15,   1/30,  -1/30;
    0,   -1/30,  1/15,   1/15,  -1/30,    0;
   1/42, -1/42, -1/105,  8/105, -1/105, -1/42,   1/42;
    0,    1/42, -1/21,   4/105,  4/105, -1/21,   1/42,   0;
  -1/30,  1/30, -1/105, -4/105,  8/105, -4/105, -1/105, 1/30, -1/30;

Links

Crossrefs

Cf. A085738, A212196. See A051714/A051715 for another triangle that generates the Bernoulli numbers.

Programs

  • Maple
    nmax:=11; for n from 0 to nmax do T(n, 0):= (-1)^n*bernoulli(n) od: for n from 1 to nmax do for k from 1 to n do  T(n, k) := T(n-1, k-1) - T(n, k-1) od: od: for n from 0 to nmax do seq(T(n, k), k=0..n) od: seq(seq(numer(T(n, k)), k=0..n), n=0..nmax);  # Johannes W. Meijer, Jun 29 2011, revised Nov 25 2012
  • Mathematica
    t[n_, 0] := (-1)^n*BernoulliB[n]; t[n_, k_] := t[n, k] = t[n-1, k-1] - t[n, k-1]; Table[t[n, k] // Numerator, {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 07 2014 *)
  • Sage
    def BernoulliDifferenceTable(n) :
    def T(S, a) :
        R = [a]
        for s in S :
            a -= s
            R.append(a)
        return R
    def M(A, p) :
        R = T(A,0)
        S = add(r for r in R)
        return -S / (2*p+3)
    R = [1/1]
    A = [1/2,-1/2]; R.extend(A)
    for k in (0..n-2) :
        A = T(A,M(A,k)); R.extend(A)
        A = T(A,0); R.extend(A)
    return R
def A085737_list(n) : return [numerator(q) for q in BernoulliDifferenceTable(n)]
# Peter Luschny, May 04 2012

Formula

T(n, 0) = (-1)^n*Bernoulli(n), T(n, k) = T(n-1, k-1) - T(n, k-1) for k=1..n.
T(n,k) = Sum_{j=0..k} binomial(k,j)*Bernoulli(n-j). [Lange and Grabisch]

Extensions

Sign flipped in formula by Johannes W. Meijer, Jun 29 2011

A291447 Triangle read by rows, numerators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = BernoulliMedian(n).

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, 0, 0, 1, -1, 4, 0, 0, 0, 1, -3, 48, -12, 36, 0, 0, 0, 1, -7, 268, -176, 1968, -216, 64, 0, 0, 0, 1, -15, 240, -1580, 37140, -9900, 10400, -5760, 14400, 0, 0, 0, 1, -31, 4924, -11680, 488640, -238680, 496320, -639360, 5486400, -216000, 518400
Offset: 0

Views

Author

Peter Luschny, Aug 24 2017

Keywords

Comments

The Bernoulli median numbers are A212196/A181131. See A290694 for further comments.

Examples

			Triangle starts:
[0, 1]
[0, 0, 0, 1]
[0, 0, 0, 1, -1, 4]
[0, 0, 0, 1, -3, 48, -12, 36]
[0, 0, 0, 1, -7, 268, -176, 1968, -216, 64]
[0, 0, 0, 1, -15, 240, -1580, 37140, -9900, 10400, -5760, 14400]
The first few polynomials are:
P_0(x) = x.
P_1(x) = (1/3)*x^3.
P_2(x) = (4/5)*x^5 - x^4 + (1/3)*x^3.
P_3(x) = (36/7)*x^7 - 12*x^6 + (48/5)*x^5 - 3*x^4 + (1/3)*x^3.
P_4(x) = 64*x^9 - 216*x^8 + (1968/7)*x^7 - 176*x^6 + (268/5)*x^5 - 7*x^4 +(1/3)*x^3.
Evaluated at x = 1 this gives a decomposition of the Bernoulli median numbers:
BM(0) = 1     =    1.
BM(1) = 1/3   =  1/3.
BM(2) = 2/15  =  4/5 -   1 +    1/3.
BM(3) = 8/105 = 36/7 -  12 +   48/5 -   3 +   1/3.
BM(4) = 8/105 =   64 - 216 + 1968/7 - 176 + 268/5 - 7 + 1/3.

Links

Crossrefs

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

Programs

  • Maple
    # The function BG_row is defined in A290694.
seq(BG_row(2, n, "num", "val"), n=0..12);        # A212196
seq(BG_row(2, n, "den", "val"), n=0..12);        # A181131
seq(print(BG_row(2, n, "num", "poly")), n=0..7); # A291447 (this seq.)
seq(print(BG_row(2, n, "den", "poly")), n=0..9); # A291448
  • Mathematica
    T[n_] := Integrate[Sum[(-1)^(n-j+1) StirlingS2[n, j] j! x^j, {j, 0, n}]^2, x];
Trow[n_] := CoefficientList[T[n], x] // Numerator;
Table[Trow[r], {r, 0, 6}] // Flatten

Formula

T(n,k) = Numerator([x^k] Integral(Sum_{j=0..n}(-1)^(n-j)*Stirling2(n,j)*j!*x^j)^m) for m = 2, n >= 0 and k = 0..m*n+1.

A291448 Triangle read by rows, denominators of coefficients (in rising powers) of rational polynomials P(n,x) such that Integral_{x=0..1} P'(n,x) = BernoulliMedian(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 5, 1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 11, 1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 11, 1, 13, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 11, 1, 13, 1, 1
Offset: 0

Views

Author

Peter Luschny, Aug 24 2017

Keywords

Comments

See A291447 and A290694 for comments.

Examples

			Triangle starts:
[1, 1]
[1, 1, 1, 3]
[1, 1, 1, 3, 1, 5]
[1, 1, 1, 3, 1, 5, 1, 7]
[1, 1, 1, 3, 1, 5, 1, 7, 1, 1]
[1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 11]
[1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13]

Crossrefs

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

Programs

  • Maple
    # See A291447.
  • Mathematica
    T[n_] := Integrate[Sum[(-1)^(n-j+1) StirlingS2[n, j] j! x^j, {j,0,n}]^2, x];
Trow[n_] := CoefficientList[T[n], x] // Denominator;
Table[Trow[r], {r, 0, 7}] // Flatten

Formula

T(n,k) = Denominator([x^k] Integral(Sum_{j=0..n}(-1)^(n-j)*Stirling2(n,j)*j!*x^j)^m) for m = 2, n >= 0 and k = 0..m*n+1.

A290694 Triangle read by rows, numerators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = Bernoulli(n, 1).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, -1, 2, 0, 0, 1, -2, 3, 0, 0, -1, 14, -9, 24, 0, 0, 1, -10, 75, -48, 20, 0, 0, -1, 62, -135, 312, -300, 720, 0, 0, 1, -42, 903, -1680, 2800, -2160, 630, 0, 0, -1, 254, -1449, 40824, -21000, 27360, -17640, 4480
Offset: 0

Views

Author

Peter Luschny, Aug 24 2017

Keywords

Comments

Consider a family of integrals I_m(n) = Integral_{x=0..1} P'(n, x)^m with P'(n,x) = Sum_{k=0..n}(-1)^(n-k)*Stirling2(n, k)*k!*x^k (see A278075 for the coefficients).
I_1(n) are the Bernoulli numbers A164555/A027642, I_2(n) are the Bernoulli median numbers A212196/A181131, I_3(n) are the numbers A291449/A291450.
The coefficients of the polynomials P_n(x)^m are for m = 1 A290694/A290695 and for m = 2 A291447/A291448.
Only omega(Clausen(n)) = A001221(A160014(n,1)) = A067513(n) coefficients are rational numbers if n is even. For odd n > 1 there are two rational coefficients.
Let C_k(n) = [x^k] P_n(x), k > 0 and n even. Conjecture: k is a prime factor of Clausen(n) <=> k = denominator(C_k(n)) <=> k does not divide Stirling2(n, k-1)*(k-1)!. (Note that by a comment in A019538 Stirling2(n, k-1)*(k-1)! is the number of chain topologies on an n-set having k open sets.)

Examples

			Triangle starts:
[0, 1]
[0, 0,  1]
[0, 0, -1,   2]
[0, 0,  1,  -2,    3]
[0, 0, -1,  14,   -9,  24]
[0, 0,  1, -10,   75, -48,   20]
[0, 0, -1,  62, -135, 312, -300, 720]
The first few polynomials are:
P_0(x) = x.
P_1(x) =  (1/2)*x^2.
P_2(x) = -(1/2)*x^2 + (2/3)*x^3.
P_3(x) =  (1/2)*x^2 - 2*x^3 + (3/2)*x^4.
P_4(x) = -(1/2)*x^2 + (14/3)*x^3 - 9*x^4 + (24/5)*x^5.
P_5(x) =  (1/2)*x^2 - 10*x^3 + (75/2)*x^4 - 48*x^5 + 20*x^6.
P_6(x) = -(1/2)*x^2 + (62/3)*x^3 - 135*x^4 + 312*x^5 - 300*x^6 + (720/7)*x^7.
Evaluated at x = 1 this gives an additive decomposition of the Bernoulli numbers:
B(0) =     1 =    1.
B(1) =   1/2 =  1/2.
B(2) =   1/6 = -1/2 +  2/3.
B(3) =     0 =  1/2 -    2 + 3/2.
B(4) = -1/30 = -1/2 + 14/3 -    9 + 24/5.
B(5) =     0 =  1/2 -   10 + 75/2 -   48 +  20.
B(6) =  1/42 = -1/2 + 62/3 -  135 +  312 - 300 + 720/7.

Links

Crossrefs

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.
Cf. A160014, A278075.

Programs

  • Maple
    BG_row := proc(m, n, frac, val) local F, g, v;
F := (n, x) -> add((-1)^(n-k)*Stirling2(n,k)*k!*x^k, k=0..n):
g := x -> int(F(n,x)^m, x):
`if`(val = "val", subs(x=1, g(x)), [seq(coeff(g(x),x,j), j=0..m*n+1)]):
`if`(frac = "num", numer(%), denom(%)) end:
seq(BG_row(1, n, "num", "val"), n=0..16);         # A164555
seq(BG_row(1, n, "den", "val"), n=0..16);         # A027642
seq(print(BG_row(1, n, "num", "poly")), n=0..12); # A290694 (this seq.)
seq(print(BG_row(1, n, "den", "poly")), n=0..12); # A290695
# Alternatively:
T_row := n -> numer(PolynomialTools:-CoefficientList(add((-1)^(n-j+1)*Stirling2(n, j-1)*(j-1)!*x^j/j, j=1..n+1), x)): for n from 0 to 6 do T_row(n) od;
  • Mathematica
    T[n_, k_] := If[k > 0, Numerator[StirlingS2[n, k - 1]*(k - 1)! / k], 0]; Table[T[n, k], {n, 0, 8}, {k, 0, n+1}] // Flatten

Formula

T(n, k) = Numerator(Stirling2(n, k - 1)*(k - 1)!/k) if k > 0 else 0; for n >= 0 and 0 <= k <= n+1.

A290695 Triangle read by rows, denominators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = Bernoulli(n, 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 5, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 7, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 5, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 0

Views

Author

Peter Luschny, Aug 24 2017

Keywords

Comments

See A290694 for comments.

Examples

			Triangle starts:
[1, 1]
[1, 1, 2]
[1, 1, 2, 3]
[1, 1, 2, 1, 2]
[1, 1, 2, 3, 1, 5]
[1, 1, 2, 1, 2, 1, 1]
[1, 1, 2, 3, 1, 1, 1, 7]
[1, 1, 2, 1, 2, 1, 1, 1, 1]

Crossrefs

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

Programs

  • Maple
    T_row := n -> denom(PolynomialTools:-CoefficientList(add((-1)^(n-j+1)*Stirling2(n, j-1)*(j-1)!*x^j/j, j=1..n+1), x)): for n from 0 to 7 do T_row(n) od;
  • Mathematica
    T[n_] := Denominator[CoefficientList[Sum[(-1)^(n-j+1) StirlingS2[n, j-1] (j-1)! x^j/j, {j, 1, n+1}], x]];
Table[T[n], {n, 0, 7}] (* Jean-François Alcover, Jun 15 2019, from Maple *)

Formula

T(n, k) = Denominator([x^k] Integral (Sum_{j=0..n} (-1)^(n-j)*Stirling2(n,j)*j!* x^j)^m) for m = 1 and k = 0..n+1.

A291449 Numerators of Integral_{x=0..1} P(n, x)^3 with P(n, x) = Sum_{k=0..n} (-1)^(n-k)* Stirling2(n, k)*k!*x^k.

Original entry on oeis.org

1, 1, 13, 1, 43, -61, 728877, 81739, -1779449713, -2112052153, 730622680308569, 113221320488699, -3660430816956396309, -3021604582205161, 21842539561810574341396283, 66747470298418575790593659, -124586733960451680357554181608419, -28471605423890788373026535240299
Offset: 0

Views

Author

Peter Luschny, Aug 24 2017

Keywords

Comments

Consider a family of integrals I(m, n) = Integral_{x=0..1} P(n, x)^m with P(n, x) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*k!*x^k. I(1, n) are the Bernoulli numbers A164555/A027642, I(2, n) are the Bernoulli median numbers A212196/A181131, I(3, n) are the numbers A291449/A291450. The coefficients of the polynomials P(n, x)^m are for m = 1 A290694/A290695, for m = 2 A291447/A291448. (See A290694 for further comments.)

Crossrefs

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

Programs

  • Maple
    # Function BG_row is defined in A290694.
seq(BG_row(3, n, "num", "val"), n=0..17);
  • Mathematica
    P[n_, x_] := Sum[(-1)^(n-k)*StirlingS2[n, k]*k!*x^k, {k, 0, n}];
a[n_] := Integrate[P[n, x]^3, {x, 0, 1}] // Numerator;
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jun 15 2019 *)

A291450 Denominators of Integral_{x=0..1} P(n, x)^3 with P(n, x) = Sum_{k=0..n}(-1)^(n-k)* Stirling2(n, k)*k!*x^k.

Original entry on oeis.org

1, 4, 140, 28, 20020, 4004, 6466460, 184756, 148728580, 29745716, 133706993420, 2431036244, 449741705140, 31885268, 670910837521540, 134182167504308, 409926521725660940, 4822664961478364, 1278006214791766460, 1921813856829724, 242081282475556183660, 4401477863191930612
Offset: 0

Views

Author

Peter Luschny, Aug 24 2017

Keywords

Comments

See A291449 and A290694 for comments.

Crossrefs

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

Programs

  • Maple
    # Function BG_row is defined in A290694.
seq(BG_row(3, n, "den", "val"), n=0..20);
  • Mathematica
    P[n_, x_] := Sum[(-1)^(n-k)*StirlingS2[n, k]*k!*x^k, {k, 0, n}];
a[n_] := Integrate[P[n, x]^3, {x, 0, 1}] // Denominator;
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jun 15 2019 *)
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