A075264 -id:A075264 - cp's OEIS frontend

A053657 a(n) = Product_{p prime} p^{ Sum_{k>=0} floor[(n-1)/((p-1)p^k)]}.

1, 2, 24, 48, 5760, 11520, 2903040, 5806080, 1393459200, 2786918400, 367873228800, 735746457600, 24103053950976000, 48206107901952000, 578473294823424000, 1156946589646848000, 9440684171518279680000, 18881368343036559360000, 271211974879377138647040000

Offset: 1

Jean-Luc Chabert, Feb 16 2000

LCM of denominators of the coefficients of x^n*z^k in {-log(1-x)/x}^z as k=0..n, as described by triangle A075264.
Denominators of integer-valued polynomials on prime numbers (with degree n): 1/a(n) is a generator of the ideal formed by the leading coefficients of integer-valued polynomials on prime numbers with degree less than or equal to n.
Also the least common multiple of the orders of all finite subgroups of GL_n(Q) [Minkowski]. Schur's notation for the sequence is M_n = a(n+1). - Martin Lorenz (lorenz(AT)math.temple.edu), May 18 2005
This sequence also occurs in algebraic topology where it gives the denominators of the Laurent polynomials forming a regular basis for K*K, the hopf algebroid of stable cooperations for complex K-theory. Several different equivalent formulas for the terms of the sequence occur in the literature. An early reference is K. Johnson, Illinois J. Math. 28(1), 1984, pp.57-63 where it occurs in lines 1-5, page 58. A summary of some of the other formulas is given in the appendix to K. Johnson, Jour. of K-theory 2(1), 2008, 123-145. - Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008
a(n) is divisible by n!, by Legendre's formula for the highest power of a prime that divides n!. Also, a(n) is divisible by (n+1)! if and only if n+1 is not prime. - Jonathan Sondow, Jul 23 2009
Triangle A163940 is related to the divergent series 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ... for m =>-1. The left hand columns of this triangle can be generated with the MC polynomials, see A163972. The Minkowski numbers appear in the denominators of these polynomials. - Johannes W. Meijer, Oct 16 2009
Unsigned Stirling numbers of the first kind as [s + k, k] (Karamata's notation) where k = {0, 1, 2, ...} and s is in general complex results in Pochhammer[s,k]*(integer coefficient polynomial of (k-1) degree in s) / M[k], where M[k] is the least common multiple of the orders of all finite groups of n x n-matrices over rational numbers (Minkowiski's theorem) which is sequence A053657. - Lorenz H. Menke, Jr., Feb 02 2010
From Peter Bala, Feb 21 2011: (Start)
Given a subset S of the integers Z, Bhargava has shown how to associate with S a generalized factorial function, denoted n!_S, which shares many properties of the classical factorial function n!.
The present sequence is the generalized factorial function n!S associated with the set of primes S = {2,3,5,7,...}. The associated generalized exponential function E(x) = Sum{n>=1} x^(n-1)/a(n) vanishes at x = -2: i.e. Sum_{n>=1} (-2)^n/a(n) = 0.
For the table of associated generalized binomial coefficients n!_S/(k!_S*(n-k)!_S) see A186430.
This sequence is related to the Bernoulli polynomials in two ways [Chabert and Cahen]:
(1) a(n) = (n-1)!*A001898(n-1).
(2) (t/(exp(t)-1))^x = sum {n = 0..inf} P(n,x)*t^n/a(n+1),
where the P(n,x) are primitive polynomials in the ring Z[x].
If p_1,...,p_n are any n primes then the product of their pairwise differences Product_{i(End)
LCM of denominators of the coefficients of S(m+n-1,m) as polynomial in m of degree 2*(n-1), as described by triangle A202339. - Vladimir Shevelev, Dec 17 2011
Sometimes called "Minkowski numbers" (e.g., by Guralnick and Lorenz), after the German mathematician Hermann Minkowski (1864-1909). - Amiram Eldar, Aug 24 2024

			a(7)=24^3*Product_{i=1..3} A202318(i)=24^3*1*10*21=2903040. - _Vladimir Shevelev_, Dec 17 2011

Jean-Luc Chabert, Scott T. Chapman, and William W. Smith, A basis for the ring of polynomials integer-valued on prime numbers, in: Daniel Anderson (ed.), Factorization in integral domains, Lecture Notes in Pure and Appl. Math. 189, Dekker, New York, 1997.

Gheorghe Coserea, Table of n, a(n) for n = 1..541
F. Bencherif, Sur une propriété des polynômes de Stirling, 26th Journées Arithmétiques, July 6-10, 2009, Université Jean Monnet, Saint-Etienne, France. [From _Jonathan Sondow_, Jul 23 2009]
M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107 (2000), 783-799.
Paul-Jean Cahen, and J. L. Chabert, What You Should Know About Integer-Valued Polynomials, The American Mathematical Monthly, 123 (No. 4, 2016), 311-337.
J.-L. Chabert, Integer-valued polynomials on prime numbers and logarithm power expansion, European J. Combinatorics 28 (2007) 754-761. [From _Jonathan Sondow_, Jul 23 2009]
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). - From _N. J. A. Sloane_, Nov 28 2012
J.-L. Chabert and P.-J. Cahen, Old problems and new questions around integer-valued polynomials and factorial sequences.
Robert M. Guralnick and Martin Lorenz, Orders of Finite Groups of Matrices, in: William Chin, James Osterburg and Declan Quinn (eds.), Groups, Rings and Algebras, Contemporary Mathematics, Vol. 420 (2006), pp. 141-161; arXiv preprint, arXiv:math/0511191 [math.GR], 2005; author's link.
K. Johnson, The action of the stable operations of complex K-theory on coefficient groups, Illinois J. Math. 28(1), 1984, pp. 57-63. [From Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008]
K. Johnson, The invariant subalgebra and anti-invariant submodule of K_*K_{(p)}, Jour. of K-theory 2(1), 2008, 123-145. [From Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008]
Hermann Minkowski, Zur Theorie der quadratischen Formen, J. Reine Angew. Math. 101 (1887), 196-202. ( = Ges. Abh., pp. 212-218, Chelsea, New York, 1967.)
Issai Schur, Über eine Klasse von endlichen Gruppen linearer Substitutionen, Sitzungsber. Preuss. Akad. Wiss. (1905), 77-91. ( = Ges. Abh., Bd. 1, pp. 128-142, Springer-Verlag, Berlin-Heidelberg-New York, 1973.)
J.-P. Serre, Bounds for the orders of the finite subgroups of G(k), Group Representation Theory (eds. M. Geck, D. Testerman, J. Thevenaz), EPFL Press, Lausanne, 2006, 405-450.
Wikipedia, Bhargava factorial.
Gregory Gerard Wojnar, Daniel Sz. Wojnar, and Leon Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017. See p. 14.

Cf. A002207, A075264, A075266, A075267.
a(n) = n!*A163176(n). - Jonathan Sondow, Jul 23 2009
Cf. A202318.
Appears in A163972. - Johannes W. Meijer, Oct 16 2009
Cf. A001898, A007814, A163940, A186430, A202339, A212429, A346046.

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: # Peter Luschny, Jul 26 2009
ser := series((y/(exp(y)-1))^x, y, 20): a := n -> denom(coeff(ser, y, n-1)):
seq(a(n), n=1..19); # Peter Luschny, May 13 2019

m = 16; s = Expand[Normal[Series[(-Log[1-x]/x)^z, {x, 0, m}]]];
a[n_, k_] := Denominator[ Coefficient[s, x^n*z^k]];
Prepend[Apply[LCM, Table[a[n,k], {n,m}, {k,n}], {1}], 1]
(* Jean-François Alcover, May 31 2011 *)
a[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k, 0, n}], {p, Prime[ Range[n] ]}]; Array[a, 30] (* Jean-François Alcover, Nov 22 2016 *)

{a(n)=local(X=x+x^2*O(x^n),D);D=1;for(j=0,n-1,D=lcm(D,denominator( polcoeff(polcoeff((-log(1-X)/x)^z+z*O(z^j),j,z),n-1,x))));return(D)} /* Paul D. Hanna, Jun 27 2005 */

{a(n)=prod(i=1,#factor(n!)~,prime(i)^sum(k=0,#binary(n), floor((n-1)/((prime(i)-1)*prime(i)^k))))} /* Paul D. Hanna, Jun 27 2005 */

S(n, p) = {
     my(acc = 0, tmp = p-1);
     while (tmp < n, acc += floor((n-1)/tmp); tmp *= p);
     return(acc);
};
a(n) = {
     my(rv = 1);
     forprime(p = 2, n, rv *= p^S(n,p));
     return(rv);
};
vector(17, i, a(i))  \\ Gheorghe Coserea, Aug 24 2015

a(2n) = 2*a(2n-1). - Jonathan Sondow, Jul 23 2009
a(2*n+1) = 24^n * Product_{i=1..n} A202318(i). - Vladimir Shevelev, Dec 17 2011
For n>=0, A007814(a(n+1)) = n+A007814(n!). - Vladimir Shevelev, Dec 28 2011
a(n) = denominator([y^(n-1)] (y/(exp(y)-1))^x). - Peter Luschny, May 13 2019
Sum_{n>=1} 1/a(n) = A346046. - Amiram Eldar, Jul 02 2023

More terms from Paul D. Hanna, Jun 27 2005

A075266 Numerator of the coefficient of x^n in log(-log(1-x)/x).

Original entry on oeis.org

0, 1, 5, 1, 251, 19, 19087, 751, 1070017, 2857, 26842253, 434293, 703604254357, 8181904909, 1166309819657, 5044289, 8092989203533249, 5026792806787, 12600467236042756559, 69028763155644023, 8136836498467582599787

Offset: 0

Paul D. Hanna, Sep 15 2002

A series with these numerators leads to Euler's constant: gamma = 1 - 1/4 - 5/72 - 1/32 - 251/14400 - 19/1728 - 19087/2540160 - ..., see references [Blagouchine] below, as well as A262235. - Iaroslav V. Blagouchine, Sep 15 2015

Robert Israel, Table of n, a(n) for n = 1..447
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.

Cf. A053657, A075264, A075267 (denominator), A262235.

S:= series(log(-log(1-x)/x),x,51):
seq(numer(coeff(S,x,j)),j=0..50); # Robert Israel, May 17 2016
# Alternative:
a := proc(n) local r; r := proc(n) option remember; if n=0 then 1 else
1 - add(r(k)/(n-k+1), k=0..n-1) fi end: numer(r(n)/(n*(n+1))) end:
seq(a(n), n=0..20); # Peter Luschny, Apr 19 2018

Numerator[ CoefficientList[ Series[ Log[ -Log[1 - x]/x], {x, 0, 20}], x]]

@cached_function
def r(n): return 1 - sum(r(k)/(n-k+1) for k in range(n)) if n > 0 else 1
def a(n: int): return numerator(r(n)/(n*(n+1))) if n > 0 else 0
print([a(n) for n in range(21)])  # Peter Luschny, Aug 15 2025

a(n) = numerator(Sum_{k=1..n} (k-1)!*(-1)^(n-k-1)*binomial(n,k)*Stirling1(n+k,k)/(n+k)!). - Vladimir Kruchinin, Aug 14 2025

Edited by Robert G. Wilson v, Sep 17 2002

a(0) = 0 prepended by Peter Luschny, Aug 15 2025

A075267 Denominator of the coefficient of x^n in log(-log(1-x)/x).

Original entry on oeis.org

2, 24, 8, 2880, 288, 362880, 17280, 29030400, 89600, 958003200, 17418240, 31384184832000, 402361344000, 62768369664000, 295206912, 512189896458240000, 342372925440000, 919636959090769920000, 5377993912811520000, 674400436666564608000000, 89903156428800000

Offset: 1

Paul D. Hanna, Sep 15 2002

Robert Israel, Table of n, a(n) for n = 1..443

Cf. A075266 (numerator), A075264, A053657.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Log(-Log(1-x)/x) )); [Denominator(b[n]): n in [1..m-2]]; // G. C. Greubel, Oct 29 2018

S:= series(log(-log(1-x)/x),x,51):
seq(denom(coeff(S,x,j)),j=1..50); # Robert Israel, May 17 2016

Denominator[ CoefficientList[ Series[ Log[ -Log[1 - x]/x], {x, 0, 18}], x]]

a(n) = denominator(Sum_{k=1..n} (k-1)!*(-1)^(n-k-1)*binomial(n,k)*Stirling1(n+k,k)/(n+k)!). - Vladimir Kruchinin, Aug 14 2025

Edited by Robert G. Wilson v, Sep 17 2002

A163972 The MC polynomials.

Original entry on oeis.org

1, 0, 3, 1, 0, 2, 45, 22, 3, 0, 0, 10, 107, 61, 13, 1, 0, -48, 20, 2100, 14855, 9168, 2390, 300, 15, 0, 0, -336, 92, 6320, 33765, 21803, 6378, 1010, 85, 3, 0, 11520, -2016, -198296, 33012, 2199246, 9547461, 6331782, 1994265, 362474, 39375, 2394, 63

Offset: 1

Johannes W. Meijer, Aug 13 2009

The a(n,p) polynomials, see below with the extra p for the column number, generate the coefficients of the left hand columns of triangle A163940. These polynomials are interesting in their own right. They have many curious properties; e.g., for p >= 1: a(n=1, p) = p, a(n=0, p) = 0, a(n = -1, p) = (-1)^(p+1), a(n=-2,p) = (-1)^(p+1)*(2)^(p-2) and a(n = -(2*p+1), 2*p) = 0, which is the outermost zero of the a(n, 2*p); for p >= 10: a(n=-10, p) = -362880*10^(p-10); etc.
The numbers in the denominators of the a(n,p) are the Minkowski numbers A053657.
The Maple program generates the coefficients of the polynomials that appear in the numerators of the a(n,p), see the sequence above. We have made use of a nice little program that Peter Luschny recently wrote for the Minkowski numbers! For the an(p,k) in the Maple program for p >= 1 we have 0 <= k <= (2*p-2). A word of caution: The value of nmax has to be chosen sufficiently large in order to let Maple find the o.g.f.s.
The zero patterns of the a(n,p) polynomials resemble the Montezuma Cypress (Taxodium mucronatum). A famous Montezuma Cypress is 'El Arbol del Tule' (the Tule tree) in Mexico. It is the second stoutest tree in the world, circumference 36 meters, and is approximately 1500 years old. Considering this I propose to call the a(n,p) polynomials the MC polynomials.
The row sums equal n*A053657(n). [Johannes W. Meijer, Nov 29 2012]

			The a(n,p) formulas of the first few left hand columns of the A163940 triangle (p is the column number):
a(n,1) = (1)/1
a(n,2) = (0 + 3*n + n^2)/2
a(n,3) = (0 + 2*n + 45*n^2+ 22*n^3 + 3*n^4)/24
a(n,4) = (0 + 0*n + 10*n^2 + 107*n^3 + 61*n^4 + 13*n^5 + n^6)/48
a(n,5) = (0 - 48*n + 20*n^2 + 2100*n^3 + 14855*n^4 + 9168*n^5 + 2390*n^6 + 300*n^7 + 15*n^8)/5760
a(n,6) = (0 + 0*n -336*n^2 +92*n^3 +6320*n^4 +33765*n^5 +21803*n^6 +6378*n^7 +1010*n^8 +85*n^9 +3*n^10)/11520
a(n,7) = (0 + 11520*n -2016*n^2 -198296*n^3 +33012*n^4 +2199246*n^5 +9547461*n^6+ 6331782*n^7 +1994265*n^8 +362474*n^9 +39375*n^10 +2394*n^11 +63*n^12)/2903040

Johannes W. Meijer, The zeros of the MC polynomials, pdf and jpg.

A000012, A000096, A163943 and A163944 are the first four left hand columns of A163940.

Cf. A053657 (Minkowski), A163402 and A075264.

pmax:=6; nmax:=70; with(genfunc): 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: for px from 1 to nmax do Gf(px):= convert(series(1/((1-(px-1)*x)^2*product((1-k*x), k=1..px-2)),x,nmax+1-px),polynom): for qy from 0 to nmax-px do a(px+qy,qy):=coeff(Gf(px),x,qy) od; od: for p from 1 to pmax do f(x):=0: for ny from p to nmax do f(x):=f(x)+a(ny,p-1)*x^(ny-p) od: f(x):= series(f(x),x, nmax): Gx:=convert(%, ratpoly): rgf_sequence('recur',Gx,x,G,n): a(n,p):=sort(simplify (rgf_expand(Gx,x,n)),n): f(p):=sort(a(n,p)*A053657(p),n,ascending): for k from 0 to 2*p-2 do an(p,k):= coeff(f(p),n,k) od; od: T:=1: for p from 1 to pmax do for k from 0 to 2*p-2 do a(T):=an(p,k): T:=T+1 od: od: seq(a(n),n=1..T-1); for p from 1 to pmax do seq(an(p,k),k=0..2*p-2) od; for p from 1 to pmax do MC(n,p):=sort(a(n,p),n,ascending) od;

A202339 Triangle of numerators of coefficients of the polynomial Q_m(n) defined by the recursion Q_0(n)=1; for m >= 1, Q_m(n) = Sum_{i=1..n} iQ_(m-1)(i). For m >= 1, the denominator for all 2m+1 terms of the m-th row is A053657(m+1).

Original entry on oeis.org

1, 1, 1, 0, 3, 10, 9, 2, 0, 1, 7, 17, 17, 6, 0, 0, 15, 180, 830, 1848, 2015, 900, 20, 0, -48, 3, 55, 410, 1598, 3467, 4055, 2120, 52, -240, 0, 0, 63, 1638, 17955, 107954, 387009, 837426, 1038681, 606606, 9828, -113624, -2016, 11520, 0, 9, 315, 4767, 40859, 217973, 747021, 1628877, 2122953, 1344798, -5516, -374024, -2592, 80640, 0, 0

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Dec 17 2011

For the first term c(m) of the m-th row, we have c(m) = A053657(m)/(2*m-2)!!.

			Q_0 = 1,
Q_1 = (x^2 + x)/2,
Q_2 = (3x^4 + 10x^3 + 9x^2 + 2x)/24,
Q_3 = (x^6 + 7x^5 + 17x^4 + 17x^3 + 6x^2)/48,
Q_4 = (15x^8 + 180x^7 + 830x^6 + 1848x^5 + 2015x^4 + 900x^3 + 20x^2 -48x)/5760,
Q_5 = (3x^10 + 55x^9 + 410x^8 + 1598x^7 + 3467x^6 + 4055x^5 + 2120x^4 + 52x^3 -240x^2)/11520,
Q_6 = (63x^12 + 1638x^11 + 17955x^10 + 107954x^9 + 387009x^8 + 837426x^7 + 1038681x^6 + 606606x^5 + 9828x^4 -113624x^3 -2016x^2 + 11520x)/2903040,
Q_7 = (9x^14 + 315x^13 + 4767x^12 + 40859x^11 + 217973x^10 + 747021x^9 + 1628877x^8 + 2122953x^7 + 1344798x^6 -5516x^5 -374024x^4 -2592x^3 + 80640x^2)/5806080,
Q_8 = (135x^16 + 6120x^15 + 122220x*14 + 1414560x^13 + 10493770x^12 + 52032240x^11 + 173988644x^10 + 384104160x^9 + 522150135x^8 + 351312360x^7 -13192648x^6 -135368640x^5 + 2658160x^4 + 49034880x^3 + 509184x^2 -5806080x)/1393459200.

Norman Do and Paul Norbury, Pruned Hurwitz numbers, arXiv preprint arXiv:1312.7516 [math.GT], 2013.

Cf. A075264, A053657, A163972.

A053657[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k, 0, n}], {p, Prime[Range[n]]}]; Q[0, n_] = 1; Q[m_, n_] := Q[m, n] = Sum[i*Q[m-1, i], {i, 1, n}]; Table[A053657[m+1]*CoefficientList[Q[m, n], n] // Reverse, {m, 0, 7}] // Flatten (* Jean-François Alcover, Nov 22 2016 *)

Q_m(n) = S(n+m, n), where S(k,l) are Stirling numbers of the second kind.

In particular, since S(m+1,1)=1, then Q_m(1)=1.

A075263 Triangle of coefficients of polynomials H(n,x) formed from the first (n+1) terms of the power series expansion of ( -x/log(1-x) )^(n+1), multiplied by n!.

Original entry on oeis.org

1, 1, -1, 2, -3, 1, 6, -12, 7, -1, 24, -60, 50, -15, 1, 120, -360, 390, -180, 31, -1, 720, -2520, 3360, -2100, 602, -63, 1, 5040, -20160, 31920, -25200, 10206, -1932, 127, -1, 40320, -181440, 332640, -317520, 166824, -46620, 6050, -255, 1, 362880, -1814400, 3780000, -4233600, 2739240, -1020600, 204630, -18660, 511, -1

Offset: 0

Paul D. Hanna, Sep 13 2002

Special values: H(n,1)=0, H(2n,2)=0, H(n,-x) ~= ( x/log(1+x) )^(n+1), for x>0. H'(n,1) = -1/n!, where H'(n,x) = d/dx H(n,x).
The zeros of these polynomials are all positive reals >= 1. If we order the zeros of H(n,x), {r_k, k=0..(n-1)}, by magnitude so that r_0 = 1, r_k > r_(k-1), for 0 < k < n, then r_(n-k) = r_k/(r_k - 1) when 0 < k < n, n > 1, where r_(n/2) = 2 for even n.
Also Product_{k=0..(n-1)} r_k = n!, r_(n-1) ~ C 2^n.
I believe that these numbers are the coefficients of the Eulerian polynomials An(z) written in powers of z-1. That is, the sequence is: A0(1); A1(1), A1'(1); A2(1), A2'(1), A2''(1)/2!; A3(1), A3'(1), A3''(1)/2!, A3'''(1)/3!; A4(1), A4'(1), A4''(1)/2!, A4'''(1)/3!, A4''''(1)/4! etc. My convention: A0(z)=z, A1(z)=z, A2(z)=z+z^2, A3(z)=z+4z^2+z^3, A4(z)=z+11z^2+11z^3+z^4, etc. - Louis Zulli (zullil(AT)lafayette.edu), Jan 19 2005
H(n,2) gives 1,-1,0,2,0,-16,0,272,0,-7936,0,..., see A009006. - Philippe Deléham, Aug 20 2007
Row sums are zero except for first row. - Roger L. Bagula, Sep 11 2008
From Groux Roland, May 12 2011: (Start)
Let f(x) = (exp(x)+1)^(-1) then the n-th derivative of f equals Sum_{k=0..n} T(n,k)*(f(x))^(n+1-k).
T(n+1,0) = (n+1)*T(n,0); T(n+1,n+1) = -T(n,n) and for 0 < k < n T(n+1,k) = (n+1-k) * T(n,k) - (n-k+2)*T(n,k-1).
T(n,k) = Sum_{i=0..k} (-1)^(i+k)*(n-i)!*binomial(n-i,k-i)*S(n,n-i) where S(n,k) is a Stirling number of the second kind. (End)

			H(0,x) = 1
H(1,x) = (1 - 1*x)/1!
H(2,x) = (2 - 3*x + 1*x^2)/2!
H(3,x) = (6 - 12*x + 7*x^2 - 1*x^3)/3!
H(4,x) = (24 - 60*x + 50*x^2 - 15*x^3 + 1*x^4)/4!
H(5,x) = (120 - 360*x + 390*x^2 - 180*x^3 + 31*x^4 - 1*x^5)/5!
H(6,x) = (720 - 2520*x + 3360*x^2 - 2100*x^3 + 602*x^4 - 63*x^5 + 1*x^5)/6!
Triangle begins:
     1;
     1,     -1;
     2,     -3,     1;
     6,    -12,     7,     -1;
    24,    -60,    50,    -15,     1;
   120,   -360,   390,   -180,    31,    -1;
   720,  -2520,  3360,  -2100,   602,   -63,   1;
  5040, -20160, 31920, -25200, 10206, -1932, 127, -1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Nguyen-Huu-Bong, Some Combinatorial Properties of Summation Operators, J. Comb. Theory, Ser. A 11.3 (1971): 213-221. See Table on page 214.

Cf. A028246, A075264, A084938, A123125.

Cf. Eulerian numbers (A008292).

Flat(List([0..12], n-> List([0..n], k-> Sum([0..n-k], j->
(-1)^(n-j)*Binomial(n-k,j)*(j+1)^n )))); # G. C. Greubel, Jan 27 2020

T:= func< n,k | &+[(-1)^(n-j)*Binomial(n-k,j)*(j+1)^n: j in [0..n-k]] >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 27 2020

CL := f -> PolynomialTools:-CoefficientList(f,x):
T_row := n -> `if`(n=0, [1], CL(x^(n+1)*polylog(-n, 1-x))):
for n from 0 to 6 do T_row(n) od; # Peter Luschny, Sep 28 2017

Table[CoefficientList[x^(n+1)*Sum[k^n*(1-x)^k, {k, 0, Infinity}], x], {n, 0, 10}]//Flatten (* Roger L. Bagula, Sep 11 2008 *)
p[x_, n_]:= x^(n+1)*PolyLog[-n, 1-x]; Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten (* Roger L. Bagula and Gary W. Adamson, Sep 15 2008 *)

T(n,k)=if(k<0 || k>n,0,n!*polcoeff((-x/log(1-x+x^2*O(x^n)))^(n+1),k))
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

T(n,k)=sum(i=0,n-k,(-1)^(n-i)*binomial(n-k,i)*(i+1)^n)
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* Using e.g.f. A(x,y): */
{T(n,k)=local(X=x+x*O(x^n),Y=y+y^2*O(y^(k))); n!*polcoeff(polcoeff(-log(1-(1-exp(-X*Y))/y),n,x),k,y)}
for(n=0,10,for(k=0,n-1,print1(T(n,k),", "));print(""))

/* Deléham's DELTA: T(n,k) = [x^(n-k)*y^k] P(n,0) */
{P(n,k)=if(n<0||k<0,0,if(n==0,1, P(n,k-1)+(x*(k\2+1)+y*(-(k\2+1)*((k+1)%2)))*P(n-1,k+1)))}
{T(n,k)=polcoeff(polcoeff(P(n,0),n-k,x),k,y)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

def T(n, k): return sum( (-1)^(n-j)*binomial(n-k, j)*(j+1)^n for j in (0..n-k))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 27 2020

Generated by [1, 1, 2, 2, 3, 3, ...] DELTA [ -1, 0, -2, 0, -3, 0, ...], where DELTA is the operator defined in A084938.
T(n, k) = Sum_{i=0..n-k} (-1)^(n-i)*C(n-k, i)*(i+1)^n; n >= 0, 0 <= k <= n. - Paul D. Hanna, Jul 21 2005
E.g.f.: A(x, y) = -log(1-(1-exp(-x*y))/y). - Paul D. Hanna, Jul 21 2005
p(x,n) = x^(n + 1)*Sum_{k>=0} k^n*(1 - x)^k; t(n,m) = Coefficients(p(x,n)). - Roger L. Bagula, Sep 11 2008
p(x,n) = x^(n + 1)*PolyLog(-n, 1 - x); t(n,m) = coefficients(p(x,n)) for n >= 1. - Roger L. Bagula and Gary W. Adamson, Sep 15 2008

Error in one term corrected by Benoit Cloitre, Aug 20 2007

A255616 Table read by antidiagonals, T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 4, 5, 4, 1, 1, 2, 5, 8, 9, 5, 1, 1, 2, 6, 11, 16, 15, 8, 1, 1, 2, 7, 14, 25, 32, 27, 11, 1, 1, 3, 8, 18, 36, 55, 64, 46, 16, 1, 1, 3, 9, 22, 49, 88, 125, 128, 81, 22, 1, 1, 3, 10, 27, 64, 129, 216, 279, 256, 140, 32, 1, 1, 3, 11, 31, 81, 181, 343, 529, 625, 512, 243, 45, 1

Offset: 0

Kival Ngaokrajang, Feb 28 2015

			See table in the links.

G. C. Greubel, Table of n, a(n) for the first 100 antidaigonals, flattened
Kival Ngaokrajang, Example of table T(n,k), n = 0..12, k = 1..10

Rows(1..10): A000012, A000196, A000027, A000093, A000290, A155013, A000578, A155014, A000583, A238170.
Columns(1..10): A000012, A017910, A017913, A000079, A017919, A017922, A017925, A017928, A000244, A017934.
Diagonals: A190343, A066642, A075264.

T[n_, k_] := Floor[Sqrt[k^n]]; Table[T[k, n + 1 - k], {n, 0, 15}, {k, 0, n}] (* G. C. Greubel, Dec 30 2017 *)

{for(i=1,20,for(n=0,i-1,a=floor(sqrt((i-n)^n));print1(a,", ")))}

T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

Terms a(81) onward added by G. C. Greubel, Dec 30 2017

Showing 1-7 of 7 results.

cp's OEIS Frontend

A053657 a(n) = Product_{p prime} p^{ Sum_{k>=0} floor[(n-1)/((p-1)p^k)]}.

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A075266 Numerator of the coefficient of x^n in log(-log(1-x)/x).

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A075267 Denominator of the coefficient of x^n in log(-log(1-x)/x).

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A163972 The MC polynomials.

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A202339 Triangle of numerators of coefficients of the polynomial Q_m(n) defined by the recursion Q_0(n)=1; for m >= 1, Q_m(n) = Sum_{i=1..n} i*Q_(m-1)(i). For m >= 1, the denominator for all 2*m+1 terms of the m-th row is A053657(m+1).

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A075263 Triangle of coefficients of polynomials H(n,x) formed from the first (n+1) terms of the power series expansion of ( -x/log(1-x) )^(n+1), multiplied by n!.

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A202339 Triangle of numerators of coefficients of the polynomial Q_m(n) defined by the recursion Q_0(n)=1; for m >= 1, Q_m(n) = Sum_{i=1..n} iQ_(m-1)(i). For m >= 1, the denominator for all 2m+1 terms of the m-th row is A053657(m+1).