A100616 -id:A100616 - cp's OEIS frontend

A100615 Let N(n)(x) be the Nørlund polynomials as defined in A001898, with N(n)(1) equal to the usual Bernoulli numbers A027641/A027642. Sequence gives numerators of N(n)(2).

1, -1, 5, -1, 1, 1, -5, -1, 7, 3, -15, -5, 7601, 691, -91, -35, 3617, 3617, -745739, -43867, 3317609, 1222277, -5981591, -854513, 5436374093, 1181820455, -213827575, -76977927, 213745149261, 23749461029, -249859397004145, -8615841276005, 238988952277727, 84802531453387

Offset: 0

N. J. A. Sloane, Dec 03 2004

With the signs of A359738, the rational sequence reflects the identity B(z)^2 = (z + 1)*B(z) - z*B'(z), that goes back to Euler, where B(z) = z/(1 - e^(-z)) is the e.g.f. of the Bernoulli numbers with B(1) = 1/2. - Peter Luschny, Jan 23 2023

			1, -1, 5/6, -1/2, 1/10, 1/6, -5/42, -1/6, 7/30, 3/10, -15/22, -5/6, 7601/2730, 691/210, -91/6, -35/2, 3617/34, 3617/30, -745739/798, -43867/42, ... = A100615/A100616.

F. N. David, Probability Theory for Statistical Methods, Cambridge, 1949; see pp. 103-104. [There is an error in the recurrence for B_s^{(r)}.]

Robert Israel, Table of n, a(n) for n = 0..575
Madeline Beals-Reid, A Quadratic Relation in the Bernoulli Numbers, The Pump Journal of Undergraduate Research, 6 (2023), 29-39.

Cf. A001898, A027641, A027642, A100616, A359738.

S:= series((x/(exp(x)-1))^2, x, 41):
seq(numer(coeff(S,x,j)*j!), j=0..40); # Robert Israel, Jun 02 2015
# Second program:
a := n -> if n = 0 then 1 else numer(-n*bernoulli(n-1) - (n-1)*bernoulli(n)) fi:
seq(a(n), n = 0..33);  # Peter Luschny, May 18 2023

Table[Numerator@NorlundB[n, 2], {n, 0, 32}] (* Arkadiusz Wesolowski, Oct 22 2012 *)
Table[If[n == 0, 1, -Numerator[n*BernoulliB[n - 1] + (n - 1)*BernoulliB[n]]], {n, 0,  33}] (* Peter Luschny, May 18 2023 *)

a(n):=sum((-1)^k*k!/(k+1)*sum(binomial(n,j)*stirling2(n-j,k)*bern(j),j,0,n-k),k,0,n); /* Vladimir Kruchinin, Jun 02 2015 */

a(n) = numerator(sum(j=0, n, binomial(n,j)*bernfrac(n-j)*bernfrac(j))); \\ Michel Marcus, Mar 03 2020

E.g.f.: (x/(exp(x)-1))^2. - Vladeta Jovovic, Feb 27 2006
a(n) = numerator(Sum_{k=0..n}(-1)^k*k!/(k+1)*Sum_{j=0..n-k} C(n,j)*Stirling2(n-j,k)*B(j)), where B(n) is Bernoulli numbers. - Vladimir Kruchinin, Jun 02 2015
a(n) = numerator(Sum_{j=0..n} binomial(n,j)*Bernoulli(n-j)*Bernoulli(j)). - Fabián Pereyra, Mar 02 2020
a(n) = -numerator(n*B(n-1) + (n-1)*B(n)) for n >= 1, where B(n) = Bernoulli(n, 0). - Peter Luschny, May 18 2023

A001898 Denominators of Bernoulli polynomials B(n)(x).

Original entry on oeis.org

1, 2, 12, 8, 240, 96, 4032, 1152, 34560, 7680, 101376, 18432, 50319360, 7741440, 6635520, 884736, 451215360, 53084160, 42361159680, 4459069440, 1471492915200, 140142182400, 1758147379200, 152882380800, 417368899584000, 33389511966720, 15410543984640

Offset: 0

N. J. A. Sloane

nonn

			The Bernoulli polynomials B(0)(x) through B(6)(x) are:
1;
-(1/2)*x;
(1/12)*(3*x-1)*x;
-(1/8)*(x-1)*x^2;
(1/240)*(15*x^3-30*x^2+5*x+2)*x;
-(1/96)*(x-1)*(3*x^2-7*x-2)*x^2;
(1/4032)*(63*x^5-315*x^4+315*x^3+91*x^2-42*x-16)*x.

F. N. David, Probability Theory for Statistical Methods, Cambridge, 1949; see pp. 103-104. [There is an error in the recurrence for B_s^{(r)}.]
N. E. Nørlund, Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin, 1924, p. 459.
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).

N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]
Index entries for sequences related to Bernoulli numbers.

Cf. A027641, A027642, A100615, A100616, A100655.

B:=bernoulli; b:=proc(s) option remember; local t; global r; if s=0 then RETURN(1); fi; expand((-r/s)*add( (-1)^t*binomial(s,t)*B(t)*b(s-t),t=1..s)); end; [seq(denom(b(n)),n=0..30)];

B[s_] := B[s] = If[s == 0, 1, (-x/s)*Sum[(-1)^t*Binomial[s, t]*
   BernoulliB[t]*B[s - t], {t, 1, s}]] // Factor;
a[n_] := If[n == 0, 1, B[n] // First // Denominator];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 24 2022 *)

These Bernoulli polynomials B(s) = B(s)(x) are defined by: B(0) = 1; B(s) = (-x/s)*Sum_{t=1..s} (-1)^t*binomial(s, t)*Bernoulli(t)*B(s-t), where Bernoulli(t) are the usual Bernoulli numbers A027641/A027642. Also B(s)(1) = Bernoulli(s).

Entry revised Dec 03 2004

A100655 Triangle read by rows giving coefficients in Bernoulli polynomials as defined in A001898, after multiplication by the common denominators A001898(n).

Original entry on oeis.org

1, 0, -1, 0, -1, 3, 0, 0, 1, -1, 0, 2, 5, -30, 15, 0, 0, -2, -5, 10, -3, 0, -16, -42, 91, 315, -315, 63, 0, 0, 16, 42, -7, -105, 63, -9, 0, 144, 404, -540, -2345, -840, 3150, -1260, 135, 0, 0, -144, -404, -100, 665, 448, -630, 180, -15, 0, -768, -2288, 2068, 11792, 8195, -8085, -8778, 6930, -1485, 99

Offset: 0

N. J. A. Sloane, Dec 05 2004

Let p(n, x) = Sum_{k=0..n} T(n, k)*x^k, then the polynomials (-1)^n*p(n; x)/x are called 'Stirling polynomials' by Knuth et al. (CMath, eq. 6.45). - Peter Luschny, Feb 05 2021

			The Bernoulli polynomials B(0)(x) through B(6)(x) are:
        1
    -(1/2)* x
    (1/12)*(3*x - 1)*x
    -(1/8)*(x-1)*x^2
   (1/240)*(15*x^3 - 30*x^2 + 5*x + 2)*x
   -(1/96)*(x-1)*(3*x^2 - 7*x - 2)*x^2
  (1/4032)*(63*x^5 - 315*x^4 + 315*x^3 + 91*x^2 - 42*x - 16)*x
Triangle of coefficients starts:
[0] [1],
[1] [0,  -1],
[2] [0,  -1,   3],
[3] [0,   0,   1,   -1],
[4] [0,   2,   5,  -30,    15],
[5] [0,   0,  -2,   -5,    10,   -3],
[6] [0, -16, -42,   91,   315, -315,   63],
[7] [0,   0,  16,   42,    -7, -105,   63,    -9],
[8] [0, 144, 404, -540, -2345, -840, 3150, -1260, 135].

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 1st ed.; Addison-Wesley, 1989, p. 257.

F. N. David, Probability Theory for Statistical Methods, Cambridge, 1949; see pp. 103-104. [There is an error in the recurrence for B_s^{(r)}.]
N. E. Nörlund, Vorlesungen ueber Differenzenrechnung Springer 1924, (p. 146).

Cf. A001898, A027641, A027642, A053657, A100615, A100616, A201637.

CoeffList := p -> op(PolynomialTools:-CoefficientList(simplify(p),x)):
E2 := (n, k) -> combinat[eulerian2](n, k): m := n -> mul(j-x, j = 1..n):
Epoly := n -> simplify(expand(add(E2(n, k)*binomial(x+k,2*n), k = 0..n)/m(n))):
poly := n -> Epoly(n)*denom(Epoly(n)):
seq(print(CoeffList(poly(n))), n = 0..8); # Peter Luschny, Feb 05 2021

row[n_] := NorlundB[n, x] // Together // Numerator // CoefficientList[#, x]&; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jul 06 2019, after Peter Luschny *)

# Formula (83), page 146 in Nörlund.
@cached_function
def NoerlundB(n, x):
    if n == 0: return 1
    return expand((-x/n)*add((-1)^k*binomial(n,k)*bernoulli(k)*NoerlundB(n-k,x) for k in (1..n)))
def A100655_row(n): return numerator(NoerlundB(n, x)).list()
[A100655_row(n) for n in (0..8)] # Peter Luschny, Jul 01 2019

E.g.f.: (y/(exp(y)-1))^x. - Vladeta Jovovic, Feb 27 2006

Let p(n, x) = (Sum_{k=0..n} E2(n, k)*binomial(x + k, 2*n))/(Product_{j=1..n} (j-x)), where E2 are the second-order Eulerian numbers (A201637), then T(n, k) = [x^k] M(n+1)*p(n, x), where M(n) are the Minkowski numbers (A053657). - Peter Luschny, Feb 05 2021

A197419 Triangle with the numerator of the coefficient [x^k] of the second order Bernoulli polynomial B_n^(2)(x) in row n, column 0<=k<=n.

Original entry on oeis.org

1, -1, 1, 5, -2, 1, -1, 5, -3, 1, 1, -2, 5, -4, 1, 1, 1, -5, 25, -5, 1, -5, 1, 3, -10, 25, -6, 1, -1, -5, 7, 7, -35, 35, -7, 1, 7, -4, -10, 28, 7, -28, 70, -8, 1, 3, 21, -6, -10, 21, 63, -42, 30, -9, 1, -15, 3, 21, -20, -25, 42, 21, -60, 75, -10, 1, -5, -15, 33, 77, -55, -55, 77, 33, -165, 275, -11, 1, 7601, -10, -45, 66, 231, -132, -110, 132, 99, -110, 55, -12, 1

Offset: 0

R. J. Mathar, Oct 14 2011

The a-th order Bernoulli polynomials are defined via the exponential generating function (t/(exp t -1))^a*exp(x*t) = sum_{n>=0} B_n^(a)(x) * t^n/n!. The current triangular array shows the coefficient [x^k] of B_n^(2)(x), i.e. the expansion coefficients in rising powers of the polynomial of x with a=2.

P(n,x) = x^n + 2*Sum_{m=0..n-1} binomial(n,m)*x^m*Sum_{k=1..n-m} stirling2(n-m,k)*stirling1(2+k,2)/((k+1)*(k+2)). - Vladimir Kruchinin, Oct 23 2011

			The table of the coefficients is
  1;
  -1,1;
  5/6,-2,1;     5/6-2x+x^2
  -1/2,5/2,-3,1;   -1/2+5x/2-3x^2+x^3
  1/10,-2,5,-4,1;
  1/6,1/2,-5,25/3,-5,1;
  -5/42,1,3/2,-10,25/2,-6,1;
  -1/6,-5/6,7/2,7/2,-35/2,35/2,-7,1;
  7/30,-4/3,-10/3,28/3,7,-28,70/3,-8,1;
  3/10,21/10,-6,-10,21,63/5,-42,30,-9,1;
  -15/22,3,21/2,-20,-25,42,21,-60,75/2,-10,1;
  -5/6,-15/2,33/2,77/2,-55,-55,77,33,-165/2,275/6,-11,1;
  7601/2730,-10,-45,66,231/2,-132,-110,132,99/2,-110,55,-12,1;

R. Dere and Y. Simsek, Bernoulli type polynomials on Umbral Algebra, arXiv:1110.1484 [math.CA], 2011.
Vladimir Kruchinin and D. Kruchinin, Application of a composition of generating functions for obtaining explicit formulas of polynomials, arXiv: 1211.0099 [math.NT], 2012.

Cf. A197420 (denominator), A100616, A100615 (column k=0).

A197419 := proc(n,k)
        local a,Bt,Bnx,o ,t,x;
        a := 2 ;
        Bt := (t/(exp(t)-1))^a*exp(x*t) ;
        Bnx := n!*coeftayl(Bt,t=0,n) ;
        coeftayl(Bnx,x=0,k) ;
        numer(%) ;
end proc:
seq(seq(A197419(n,k),k=0..n),n=0..4) ; # print row by row

t[n_, m_] := If [n == m, 1, 2*Binomial[n, m]*Sum[StirlingS2[n-m, k]*StirlingS1[2+k, 2]/((k+1)*(2+k)), {k, 1, n-m}]]; Table[t[n, m] // Numerator, {n, 0, 12}, {m, 0, n}] // Flatten (* Jean-François Alcover, Dec 12 2013, after Vladimir Kruchinin *)

T(n,m):=num(if n=m then 1 else 2*binomial(n,m)* sum(stirling2(n-m,k) *stirling1(2+k,2)/ ((k+1)*(2+k)),k,1,n-m)); /* Vladimir Kruchinin, Oct 23 2011 */

T(n,m) = 2*binomial(n,m)*Sum_{k=1..n-m} Stirling2(n-m,k)*Stirling1(2+k,2)/((k+1)*(k+2)), mVladimir Kruchinin, Oct 23 2011

cp's OEIS Frontend

A100615 Let N(n)(x) be the Nørlund polynomials as defined in A001898, with N(n)(1) equal to the usual Bernoulli numbers A027641/A027642. Sequence gives numerators of N(n)(2).

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A001898 Denominators of Bernoulli polynomials B(n)(x).

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A100655 Triangle read by rows giving coefficients in Bernoulli polynomials as defined in A001898, after multiplication by the common denominators A001898(n).

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A197419 Triangle with the numerator of the coefficient [x^k] of the second order Bernoulli polynomial B_n^(2)(x) in row n, column 0<=k<=n.

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