A053657 -id:A053657 - cp's OEIS frontend

A202318 Let (n)_p denote the exponent of prime p in the prime power factorization of n. Then a(n) is defined by the formulas a(1)=1; for n >= 2, (a(n))_2 = (n)_2, (a(n))_3 = (n)_3 and, for p >= 5, (a(n))_p = 1 + ((2n)/(p-1))_p if p-1|2*n, and (a(n))_p = 0 otherwise.

1, 10, 21, 20, 11, 2730, 1, 680, 1197, 550, 23, 5460, 1, 290, 7161, 1360, 1, 5757570, 1, 45100, 6321, 230, 47, 185640, 11, 530, 3591, 580, 59, 283933650, 1, 2720, 32361, 10, 781, 840605220, 1, 10, 1659, 1533400, 83, 23830170, 1, 40940, 408177, 470, 1, 36014160, 1, 277750, 2163, 1060, 107, 1882725390

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Dec 16 2011

nonn

a(n)=1 iff n has form 6n+-1 and, if d >= 5 is a divisor of n, then 2*d+1 is not prime. The places of 1's form sequence A045979.

If p is an odd prime and p^n is the side length of the odd leg of a primitive Pythagorean triangle (PPT) it constrains the other leg and hypotenuse to be (p^(2n)-1)/2 and (p^(2n)+1)/2 and the area to be (p^n-1)p^n(p^n+1)/4. Now consider the term (p^n-1)p^n(p^n+1): it must at least be divisible by 24 for all odd primes p because the area of a PPT is divisible by 6 (see A127922 for n=1). a(n) equals the common divisor of the term (p^n-1)p^n(p^n+1)/24 for all odd primes p. - Frank M Jackson, Dec 09 2017

			Let n=6. Since 2*6+1=13 is prime, the max p that should be considered is 13. We have
  (a(6))_2  = (a(6))_3 = 1,
  (a(6))_5  = (12/4)_5 + 1 = 1,
  (a(6))_7  = (12/6)_7 + 1 = 1,
  (a(6))_13 = (12/12)_13 + 1 = 1.
Thus a(6) = 2*3*5*7*13 = 2730.

Frank M Jackson, Table of n, a(n) for n = 1..10000

Cf. A053657, A045979, A002445, A127922.

Table[Numerator[Exp[Re[Limit[Zeta[s] (Zeta[-1]^(s - 1) - Zeta[-(2*n - 1)]^(s - 1)), s -> 1]]]], {n, 1, 54}] (* Mats Granvik, Feb 05 2016 *)
Table[(lst=Table[p=Prime[m+1]; (p^n-1)p^n(p^n+1), {m, 1, 10}]; GCD@@lst/24), {n, 1, 100}] (* Frank M Jackson, Dec 09 2017 *)
a[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k, 0, n}], {p, Prime[Range[n]]}]; Array[a[2#+1]/(24 a[2#-1]) &, 100] (* using Jean-François Alcover's program A053657 *)(* Frank M Jackson, Dec 16 2017 *)

a(n) = {my(r = 1); forprime(p=2, 2*n+1, if (p<=3, r *= p^valuation(n, p), if (! (2*n % (p-1)), r *= p^(1+valuation((2*n)/(p-1), p))););); r;} \\ Michel Marcus, Feb 06 2016

a(n) = (1/24)*b(2n+1)/b(2n-1), where b(n) = A053657(n).
a(p) = A002445(p)/6, for prime p >= 5.
a(n) = numerator of e^(real(lim_{s -> 1} (zeta(s)*(zeta(-1)^(s-1) - zeta(-(2*n-1))^(s-1))))). - Mats Granvik, Feb 05 2016
a(n) = A036283(n)/6. - Hugo Pfoertner, Dec 18 2022

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

A203484 For n>=0, let n!^(3) = A202368(n+1) and, for 0<=m<=n, C^(3)(n,m) = n!^(3)/(m!^(3)*(n-m)!^(3)). The sequence gives triangle of numbers C^(3)(n,m) with rows of length n+1.

Original entry on oeis.org

1, 1, 1, 1, 42, 1, 1, 5, 5, 1, 1, 1092, 130, 1092, 1, 1, 1, 26, 26, 1, 1, 1, 11970, 285, 62244, 285, 11970, 1, 1, 11, 3135, 627, 627, 3135, 11, 1

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jan 02 2012

Conjecture. If p is prime of the form 3*k+1, then the k-th row contains two 1's and k-1 numbers multiple of p; if p is prime of the form 3*k+2, then the (2*k+1)-th row contains two 1's and 2*k numbers multiple of p.

			Triangle begins
n/m.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1......1
.2..|..1.....42.....1
.3..|..1......5 ....5......1
.4..|..1...1092...130...1092.....1
.5..|..1......1....26.....26.....1......1
.6..|..1..11970...285..62244...285..11970....1
.7..|..1.....11..3135....627...627...3135...11.....1
.8..|

Cf. A175669, A053657, A202339, A202367, A202368, A202369, A202917, A202941.

Conjecture. A007814(C^(3)(n,m)) = A007814(C(n,m)).

A212429 a(n) is the LCM of denominators of polynomials of degree n which are integer-valued on primes together with their first divided differences.

Original entry on oeis.org

1, 1, 2, 4, 48, 96, 1152, 2304, 276480, 552960, 6635520, 13271040, 33443020800, 66886041600, 802632499200, 1605264998400, 385263599616000, 770527199232000, 194172854206464000, 388345708412928000, 512616335105064960000, 1025232670210129920000

Offset: 1

Jean-Luc Chabert, Jun 21 2012

nonn

a(n) is also the n-th Bhargava's factorial n_P^{{1}} of the set P of primes with respect to the first divided difference.

			a(5) = 48 because f(x) = (x-1)(x-2)(x-3)(x-5)(x-7)/48 satisfies f(p) and (f(p)-f(q))/(p-q) are integers for all primes p,q.

Alois P. Heinz, Table of n, a(n) for n = 1..170
M. Bhargava, On P-orderings, Integer-Valued Polynomials, and Ultrametric Analysis, J. Amer. Math. Soc., 22 (2009), 963-993.
J. L. Chabert, About polynomials whose divided differences are integer-valued on prime numbers, ICM 2012 Proceedings, vol. I, pp. 1-7. Complete proceedings (warning: file size is 26MB).

Cf. A053657.

a:= proc(n) local i, p, wp, r;
      r:=1;
      for i do p:= ithprime(i);
        wp:= p^(w(p,n-1));
        if wp=1 then break fi;
        r:= r*wp
      od; r
    end:
w:= proc(p, n) local d, k, r;
      r:= 0;
      for k from 0 do d:= floor(n/((p-1)*p^k));
        if d=0 then break fi;
        r:= r+d;
      od;
      r -t(n,p)
    end:
t:= proc(n, p) local h, q;
      q:= n/(p-1);
      for h from 0 while q>= p^h do od; h
    end:
seq (a(n), n=1..30);  # Alois P. Heinz, Jun 25 2012

a[n_] := Module[{i, p, wp, r}, r = 1; For[i = 1, True, i++, p = Prime[i]; wp = p^w[p, n - 1]; If[wp == 1, Break[]]; r = r*wp]; r];
w[p_, n_] := Module[{d, k, r}, r = 0; For[k = 0, True, k++, d = Floor[n/((p - 1)*p^k)]; If[d == 0, Break[]]; r = r + d]; r - t[n, p]];
t[n_, p_] := Module[{h, q}, q = n/(p - 1); For[h = 0, q >= p^h , h++]; h];
a /@ Range[1, 30] (* Jean-François Alcover, Oct 14 2019, after Alois P. Heinz *)

a(n) = Prod_{p prime} p^w_p(n-1) where w_p(n) = Sum_{k>=0} floor(n / ((p-1)*p^k)) - t_{p,n} and p^(t_{p,n}-1) <= n/(p-1) < p^t_{p,n}.

A260326 Common denominator of coefficients in Nörlund's polynomial D_{2n}(x).

Original entry on oeis.org

1, 3, 15, 63, 135, 99, 12285, 405, 6885, 161595, 1403325, 419175, 24877125, 229635, 528525, 26101845, 214708725, 1148175, 31479513975, 134336475, 23302211625, 513217002375, 374333754375, 50996192625, 25178013826875, 678264862275, 813304020375, 6122798191125

Offset: 0

N. J. A. Sloane, Jul 25 2015

Let phi_(D,rho) be the average value of a generic degree D monic polynomial f when evaluated at the roots of the rho-th derivative of f, expressed as a polynomial in the averaged symmetric polynomials in the roots of f [Wojnar et al., 2017]. The "last" term of phi_(D,rho) is a multiple of the product of all roots of f; the coefficient is expressible as a polynomial h_D(N) in N:=D-rho. These polynomials are of the form h_D(N) = ((-1)^D/(D-1)!)(D-N)N^chi(D)*g_D(N) where chi(D) := (1 if D is odd, 0 if D is even) and g_D(N) is a monic polynomial of degree (D-2-chi). The coefficients of the g_D(N) are polynomials in D of the form k_n(D) = (1/Q(n))(D+t(n))^delta(n)D^chi(n+1)u_n(D) where Q(n) = A053657(n), t(n):=2 ceiling(n/2)+1, delta(n):= (1 if n is odd, 2 if n is even). For odd n, the leading coefficients of u_n(D) are a((n+1)/2). - Gregory Gerard Wojnar, Jul 17 2017

Jean-François Alcover, Table of n, a(n) for n = 0..100
N. E. Nørlund, Vorlesungen ueber Differenzenrechnung Springer 1924, p. 460.
N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924; page 460 [Annotated scanned copy of pages 144-151 and 456-463]
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.

For numerators see A260327.

Cf. A053657.

# NorlundD polynomials are defined in A260327.
seq(denom(NorlundD(2*n)(x)), n=0..27); # Peter Luschny, Jul 01 2019

NorlundD[nu_, n_] := (-2)^nu NorlundB[nu, n, n/2] // Simplify;
a[n_] := Module[{nb}, nb = NorlundB[2n, x]; nb/Coefficient[nb, x, 2n] // Together // Denominator];
(* or: *)
a[n_] := (2n)! SeriesCoefficient[(z/Sin[z])^x, {z, 0, 2n}] // Normal // Together // Denominator;
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jul 01 2019 *)

{ A260326(n) = my(t, Y); Y=y+O(y^(2*n+2)); t = (2*n)! * Pol( polcoeff( exp( x * log(Y/sinh(Y)) + O(x^(n+1)) ), 2*n, y ) ); denominator(content(t)); } \\ Max Alekseyev, Jul 04 2019

E.g.f. Sum_{n>=0} D_{2n}(x) y^(2n)/(2n)! = (y/sinh(y))^x. - Max Alekseyev, Jul 04 2019

Terms a(7) and beyond from Gregory Gerard Wojnar, Jul 19 2017

a(24)-a(27) corrected by Jean-François Alcover, Jul 01 2019

A290030 Leading coefficients of numerators of Norlund's B_{nu}^(n) polynomials (Nørlund, Tafel 5, p. 459).

Original entry on oeis.org

1, -1, 3, -1, 15, -3, 63, -9, 135, -15, 99, -9, 12285, -945, 405, -27, 6885, -405, 161595, -8505, 1403325, -66825, 419175, -18225, 24877125, -995085, 229635, -8505, 528525, -18225, 26101845, -841995, 214708725, -6506325, 1148175, -32805, 31479513975, -850797675

Offset: 0

Gregory Gerard Wojnar, Jul 17 2017

sign

Let phi_(D,rho) be the average value of a generic degree D monic polynomial f when evaluated at the roots of the rho-th derivative of f, expressed as a polynomial in the averaged symmetric polynomials in the roots of f. [See arXiv:1706.08381 [math,GM], 2017.] The "last" term of phi_(D,rho) is a multiple of the product of all roots of f; the coefficient is expressible as a polynomial h_D(N) in N:=D-rho. These polynomials are of the form h_D(N)= ((-1)^D/(D-1)!)*(D-N)*N^chi(D)*g_D(N) where chi(D) := (1 if D is odd, 0 if D is even) and g_D(N) is a monic polynomial of degree (D-2-chi). The coefficients of the g_D(N) are polynomials in D of the form k_n(D)=(1/Q(n))*(D+t(n))^delta(n)*D^chi(n+1)*u_n(D) where Q(n)=A053657(n), t(n):=2 ceiling(n/2)+1, delta(n):= (1 if n is odd, 2 if n is even). The leading coefficients of u_n(D) are a(n).

Gregory Gerard Wojnar, Table of n, a(n) for n = 0..187
N. E. Nørlund, Vorlesungen ueber Differenzenrechnung Springer 1924, p. 459.
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.

Cf. A053657, A100655.

a[n_] := NorlundB[n, x] // Together // Numerator // Coefficient[#, x, n]&;
Table[a[n], {n, 0, 37}] (* Jean-François Alcover, Jun 30 2019 *)

[A100655_row(n)[n] for n in (0..37)] # Peter Luschny, Jul 01 2019

A358625 a(n) = numerator of Bernoulli(n, 1) / n for n >= 1, a(0) = 1.

Original entry on oeis.org

1, 1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -691, 0, 1, 0, -3617, 0, 43867, 0, -174611, 0, 77683, 0, -236364091, 0, 657931, 0, -3392780147, 0, 1723168255201, 0, -7709321041217, 0, 151628697551, 0, -26315271553053477373, 0, 154210205991661, 0, -261082718496449122051

Offset: 0

Peter Luschny, Dec 02 2022

The rational numbers r(n) = Bernoulli(n, 1) / n are called the 'divided Bernoulli numbers'. r(n) is a p-integer for all primes p if p - 1 does not divide n. This is sometimes called 'Adams's theorem' (Ireland and Rosen). The important Kummer congruences for the Bernoulli numbers (1851) are stated in terms of the r(n).

			Rationals: 1, 1/2, 1/12, 0, -1/120, 0, 1/252, 0, -1/240, 0, 1/132, ...
Note that a(68) = -4633713579924631067171126424027918014373353 but A120082(68) = -125235502160125163977598011460214000388469.

Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, Vol. 84, Graduate Texts in Mathematics, Springer-Verlag, 2nd edition, 1990. [Prop. 15.2.4, p. 238]

Peter Luschny, Table of n, a(n) for n = 0..300
Arnold Adelberg, Shaofang Hong and Wenli Ren, Bounds of Divided Universal Bernoulli Numbers and Universal Kummer Congruences, Proceedings of the American Mathematical Society, Vol. 136(1), 2008, p. 61-71.
Bernd C. Kellner, The structure of Bernoulli numbers, arXiv:math/0411498 [math.NT], 2004.

Cf. A001067, A342318, A120080, A120082, A120084, A120086, A079612, A075180, A060054.
Cf. A164555 / A027642.
For the denominators cf. A006953, A036283, A053657, A202318.

Concatenation([1, 1], List([2..45], n-> NumeratorRat(Bernoulli(n)/(n)))); # G. C. Greubel, Sep 19 2019

[1, 1] cat [Numerator(Bernoulli(n)/(n)): n in [2..45]]; // G. C. Greubel, Sep 19 2019

A358625 := n -> ifelse(n = 0, 1, numer(bernoulli(n, 1) / n)):
seq(A358625(n), n = 0.. 40);
# Alternative:
egf := 1 + x + log(1 - exp(-x)) - log(x): ser := series(egf, x, 42):
seq(numer(n! * coeff(ser, x, n)), n = 0..40);

Join[{1, 1}, Table[Numerator[BernoulliB[n] / n], {n, 2, 45}]]

a(n) = if (n<=1, 1, numerator(bernfrac(n)/n)); \\ Michel Marcus, Feb 24 2015

a(n) = numerator(n! * [x^n](1 + x + log(1 - exp(-x)) - log(x))).
a(n) = numerator(-zeta(1 - n)) for n >= 1.
a(n) = numerator(Euler(n-1, 1) / (2*(2^n - 1))) for n >= 1.
denominator(r(2*n)) = A006953(n) for n >= 1.
denominator(r(2*n)) / 2 = A036283(n) for n >= 1.
denominator(r(2*n)) / 12 = A202318(n) for n >= 1.
denominator(r(2*n)) = (1/2) * A053657(2*n+1) / A053657(2*n-1) for n >= 1.

A163402 A Minkowski-type generalization of the factorial function: F(n,k) with k = 2.

Original entry on oeis.org

1, 1, 1, 3, 9, 135, 1215, 2835, 127575, 229635, 3444525, 1705039875, 107417512125, 13299311025, 4189282972875, 62839244593125, 188517733779375, 336504154796184375, 9085612179496978125, 2740105260483215625

Offset: 0

Peter Luschny, Jul 26 2009

nonn

F(n,0) = n! (A000142)
F(n,1) = Minkowski(n)/n! (A163176)
F(n,2) = a(n)

			For n >= 0
F(n,0) 1, 1, 2, 6, 24, 120, 720, 5040, 40320, ...
F(n,1) 1, 1, 1, 4, 2, 48, 16, 576, 144, 3840, ...
F(n,2) 1, 1, 1, 3, 9, 135, 1215, 2835, 127575, ...
F(n,3) 1, 1, 1, 1, 1, 1, 1, 5, 1, 25, 5, 35, ...
F(n,4) 1, 1, 1, 1, 1, 5, 25, 175, 4375, 4375, ...
F(n,5) 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 49, ...

Cf. A000142, A053657, A163176.

F := proc(n,k) local L,p,i;
L := proc(n,u,r) local q,s,m; m:=n-r;
q:=u-r; s:=0; do if q>m then break fi;
s:=s+iquo(m,q); q:=q*u od; s end;
mul(p^add((-1)^i*L(n,p,i),i=0..k),
p = select(isprime,[$(k+1)..n]))^((-1)^k) end:
a(n) := n -> F(n,2);

F[n_, k_] := Module[{L, p, i}, L[n0_, u_, r_] := Module[{q, s, m}, m = n0-r; q = u-r; s = 0; While[True, If[q > m, Break[]]; s = s + Quotient[m, q]; q = q*u]; s]; Product[p^Sum[(-1)^i*L[n, p, i], {i, 0, k}], {p, Select[Range[k+1, n], PrimeQ]}]^((-1)^k)]; a[n_] := F[n, 2]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jan 15 2014, translated from Maple *)

def A163402(n):
    def L(n, u, r):
        m = n - r; q = u - r
        s = 0
        while(q <= m):
            s += m//q
            q *= u
        return s
    P = filter(is_prime, [3..n])
    return mul(p^add((-1)^i*L(n, p, i) for i in (0..2)) for p in P)
print([A163402(n) for n in range(20)]) # Peter Luschny, Mar 13 2016

P(n,k) = {p prime | k+1 <= p <= n }
L(n,p,r) = Sum_{i>=0} floor((n-r)/((p-r)*p^i))
A(n,k) = Prod_{p in P(n,k)} p^(Sum_{m=0..k} (-1)^m*L(n,p,m))
F(n,k) = A(n,k)^((-1)^k).

A178473 For n>=0, let n!^(4) = A202369(n+1) and, for 0<=m<=n, C^(4)(n,m) = n!^(4)/(m!^(4)*(n-m)!^(4)). The sequence gives triangle of numbers C^(4)(n,m) with rows of length n+1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 273, 273, 1, 1, 68, 9282, 68, 1, 1, 55, 1870, 1870, 55, 1, 1, 546, 15015, 3740, 15015, 546, 1, 1, 29, 7917, 1595, 1595, 7917, 29, 1

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jan 02 2012

Conjecture. If p is prime of the form 4*k+1, then the k-th row contains two 1's and k-1 numbers multiple of p; if p is prime of the form 4*k+3, then the (2*k+1)-th row contains two 1's and 2*k numbers multiple of p.

			Triangle begins
n/m.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1......1
.2..|..1......2......1
.3..|..1....273 ...273......1
.4..|..1.....68...9282.....68......1
.5..|..1.....55...1870...1870.....55......1
.6..|..1....546..15015...3740..15015....546....1
.7..|..1.....29...7917...1595...1595...7917...29.....1
.8..|

Cf. A175669, A053657, A202339, A202367, A202368, A202369, A202917, A202941, A203484.

Conjecture. A007814(C^(4)(n,m)) = A007814(C(n,m)).

A186431 Row sums of A186430.

Original entry on oeis.org

1, 2, 4, 26, 18, 482, 266, 6050, 3114, 21122, 10730, 22178, 11226, 4455362, 2256338, 343874, 173610, 13643522, 6869842, 690621122, 347772738, 16250361602, 8187307306, 17146915106, 8584448890, 720152334722, 365024665978, 59381983394, 29700003082

Offset: 0

Peter Bala, Feb 21 2011

Cf. A053657, A186430.

# A186431, uses program for A053657 written by Peter Luschny:
A053657 := proc(n) local P, p, q, s, r;
P := select(isprime, [$2..n]); r:=1;
for p in P do s := 0; q := p-1;
do if q > (n-1) then break fi;
s := s + iquo(n-1, q); q := q*p; od;
r := r * p^s; od; r end:
# Row sums:
a:= n-> add(A053657(n)/(A053657(k)*A053657(n-k)), k = 0..n):
seq (a(n), n = 0..22);

b[n_] := b[n] = Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k, 0, n}], {p, Prime[ Range[n]]}];
T[n_, k_] := b[n]/(b[k] b[n-k]);
a[n_] := Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Jun 26 2019 *)

a(n) = Sum_{k=0..n} A053657(n)/(A053657(k)*A053657(n-k)), with the convention that A053657(0) = 1.

cp's OEIS Frontend

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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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A203484 For n>=0, let n!^(3) = A202368(n+1) and, for 0<=m<=n, C^(3)(n,m) = n!^(3)/(m!^(3)*(n-m)!^(3)). The sequence gives triangle of numbers C^(3)(n,m) with rows of length n+1.

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A212429 a(n) is the LCM of denominators of polynomials of degree n which are integer-valued on primes together with their first divided differences.

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A260326 Common denominator of coefficients in Nörlund's polynomial D_{2n}(x).

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A290030 Leading coefficients of numerators of Norlund's B_{nu}^(n) polynomials (Nørlund, Tafel 5, p. 459).

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A358625 a(n) = numerator of Bernoulli(n, 1) / n for n >= 1, a(0) = 1.

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