A163972 -id:A163972 - 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

A163940 Triangle related to the divergent series 1^m1! - 2^m2! + 3^m3! - 4^m4! + ... for m >= -1.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 5, 3, 0, 1, 9, 17, 4, 0, 1, 14, 52, 49, 5, 0, 1, 20, 121, 246, 129, 6, 0, 1, 27, 240, 834, 1039, 321, 7, 0, 1, 35, 428, 2250, 5037, 4083, 769, 8, 0, 1, 44, 707, 5214, 18201, 27918, 15274, 1793, 9, 0, 1, 54, 1102, 10829, 54111, 133530, 145777, 55152, 4097, 10, 0

Offset: 0

Johannes W. Meijer, Aug 13 2009

The divergent series g(x,m) = Sum_{k >= 1} (-1)^(k+1)*k^m*k!*x^k, m >= -1, are related to the higher order exponential integrals E(x,m,n=1), see A163931.
Hardy, see the link below, describes how Euler came to the rather surprising conclusion that g(x,-1) = exp(1/x)*Ei(1,1/x) with Ei(1,x) = E(x,m=1,n=1). From this result it follows inmediately that g(x,0) = 1 - g(x,-1). Following in Euler's footsteps we discovered that g(x,m) = (-1)^(m) * (M(x,m)*x - ST(x,m)* Ei(1,1/x) * exp(1/x))/x^(m+1), m => -1.
So g(x=1,m) = (-1)^m*(A040027(m) - A000110 (m+1)*A073003), with A040027(m = -1) = 0. We observe that A073003 = - exp(1)*Ei(-1) is Gompertz's constant, A000110 are the Bell numbers and A040027 was published a few years ago by Gould.
The polynomial coefficients of the M(x,m) = Sum_{k = 0..m} a(m,k) * x^k, for m >= 0, lead to the triangle given above. We point out that M(x,m=-1) = 0.
The polynomial coefficients of the ST(x,m) = Sum_{k = 0..m+1} S2(m+1, k) * x^k, m >= -1, lead to the Stirling numbers of the second kind, see A106800.
The formulas that generate the coefficients of the left hand columns lead to the Minkowski numbers A053657. We have a closer look at them in A163972.
The right hand columns have simple generating functions, see the formulas. We used them in the first Maple program to generate the triangle coefficients (n >= 0 and 0 <= k <= n). The second Maple program calculates the values of g(x,m) for m >= -1, at x=1.

			The first few triangle rows are:
  [1]
  [1, 0]
  [1, 2, 0]
  [1, 5, 3, 0]
  [1, 9, 17, 4, 0]
  [1, 14, 52, 49, 5, 0]
The first few M(x,m) are:
  M(x,m=0) = 1
  M(x,m=1) = 1 + 0*x
  M(x,m=2) = 1 + 2*x + 0*x^2
  M(x,m=3) = 1 + 5*x + 3*x^2 + 0*x^3
The first few ST(x,m) are:
  ST(x,m=-1) = 1
  ST(x,m=0) = 1 + 0*x
  ST(x,m=1) = 1 + 1*x + 0*x^2
  ST(x,m=2) = 1 + 3*x + x^2 + 0*x^3
  ST(x,m=3) = 1 + 6*x + 7*x^2 + x^3 + 0*x^4
The first few g(x,m) are:
  g(x,-1) = (-1)*(- (1)*Ei(1,1/x)*exp(1/x))/x^0
  g(x,0) = (1)*((1)*x - (1)*Ei(1,1/x)*exp(1/x))/x^1
  g(x,1) = (-1)*((1)*x - (1+ x)*Ei(1,1/x)*exp(1/x))/x^2
  g(x,2) = (1)*((1+2*x)*x - (1+3*x+x^2)*Ei(1,1/x)*exp(1/x))/x^3
  g(x,3) = (-1)*((1+5*x+3*x^2)*x - (1+6*x+7*x^2+x^3)*Ei(1,1/x)*exp(1/x))/x^4

G. H. Hardy, Divergent Series, Oxford University Press, 1949. pp. 26-29 and pp. 7-8.
Maurice de Gosson, Branko Dragovich and Andrei Khrennikov, Some p-adic differential equations, (see Section 5), arxiv:math-ph/0010023, Oct 2000.

The row sums equal A040027 (Gould).
A000007, A000027, A000337, A163941 and A163942 are the first five right hand columns.
A000012, A000096, A163943 and A163944 are the first four left hand columns.
Cf. A163931, A163972, A106800 (Stirling2), A000110 (Bell), A073003 (Gompertz), A053657 (Minkowski), A014619.

nmax := 10; for p from 1 to nmax do Gf(p) := convert(series(1/((1-(p-1)*x)^2*product((1-k1*x), k1=1..p-2)), x, nmax+1-p), polynom); for q from 0 to nmax-p do a(p+q-1, q) := coeff(Gf(p), x, q) od: od: seq(seq(a(n, k), k=0..n), n=0..nmax-1);
# End program 1
nmax1:=nmax; A040027 := proc(n): if n = -1 then 0 elif n= 0 then 1 else add(binomial(n, k1-1)*A040027(n-k1), k1 = 1..n) fi: end: A000110 := proc(n) option remember; if n <= 1 then 1 else add( binomial(n-1, i) * A000110(n-1-i), i=0..n-1); fi; end: A073003 := - exp(1) * Ei(-1): for n from -1 to nmax1 do g(1, n) := (-1)^n * (A040027(n) - A000110(n+1) * A073003) od;
# End program 2

nmax = 11;
For[p = 1, p <= nmax, p++, gf = 1/((1-(p-1)*x)^2*Product[(1-k1*x), {k1, 1, p-2}]) + O[x]^(nmax-p+1) // Normal; For[q = 0, q <= nmax-p, q++, a[p+q-1, q] = Coefficient[gf, x, q]]];
Table[a[n, k], {n, 0, nmax-1}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 02 2019, from 1st Maple program *)

The generating functions of the right hand columns are Gf(p, x) = 1/((1 - (p-1)*x)^2 * Product_{k = 1..p-2} (1-k*x) ); Gf(1, x) = 1. For the first right hand column p = 1, for the second p = 2, etc..
From Peter Bala, Jul 23 2013: (Start)
Conjectural explicit formula: T(n,k) = Stirling2(n,n-k) + (n-k)*Sum_{j = 0..k-1} (-1)^j*Stirling2(n, n+1+j-k)*j! for 0 <= k <= n.
The n-th row polynomial R(n,x) appears to satisfy the recurrence equation R(n,x) = n*x^(n-1) + Sum_{k = 1..n-1} binomial(n,k+1)*x^(n-k-1)*R(k,x). The row polynomials appear to have only real zeros. (End)

Edited by Johannes W. Meijer, Sep 23 2012

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.

A075264 Triangle of numerators of coefficients, where the n-th row forms the polynomial in z, P(n,z), that is the coefficient of x^n in {-log(1-x)/x}^z, for n > 0. The denominator for all the terms in the n-th row is A053657(n).

Original entry on oeis.org

1, 5, 3, 6, 5, 1, 502, 485, 150, 15, 760, 802, 305, 50, 3, 152696, 171150, 73801, 15435, 1575, 63, 252336, 295748, 139020, 33817, 4515, 315, 9, 51360816, 62333204, 31231500, 8437975, 1334760, 124110, 6300, 135, 88864128, 110941776, 58415444

Offset: 1

Paul D. Hanna, Sep 15 2002; revised Jun 27 2005

Each n-th row polynomial, P(n,z), has a trivial zero at z = 0; for odd rows, P(2n+1,z) also has zeros at z = -2n, z = -(2n+1), for n > 0.

			P(1,z) = z/2,
P(2,z) = (5z + 3z^2)/24,
P(3,z) = (6z + 5z^2 + z^3)/48,
P(4,z) = (502z + 485z^2 + 150z^3 + 15z^4)/5760,
P(5,z) = (760z + 802z^2 + 305z^3 + 50z^4 +3z^5)/11520,
P(6,z) = (152696z + 171150z^2 + 73801z^3 + 15435z^4 + 1575z^5
+ 63z^6)/2903040,
P(7,z) = (252336z + 295748z^2 + 139020z^3 + 33817z^4 + 4515z^5
+ 315z^6 + 9z^7)/5806080,
P(8,z) = (51360816z + 62333204z^2 + 31231500z^3 + 8437975z^4
+ 1334760z^5 + 124110z^6 + 6300z^7 + 135z^8)/1393459200.

Cf. A053657.

Cf. A163972 (MC polynomials).

nmax:=8; 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: f(z) := convert(series((-ln(1-x)/x)^z, x, nmax+2), polynom): for n from 1 to nmax do f(n) := A053657(n+1)*coeff(f(z), x, n) od: for n from 1 to nmax do for m from 1 to n do a(n, m) := coeff(f(n), z, m) od: od: seq(seq(a(n, m), m=1..n), n=1..nmax);  # Johannes W. Meijer, Jun 08 2009, revised Nov 25 2012

rows = 9; A053657[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k, 0, n}], {p, Prime[Range[n]]}]; (Rest[CoefficientList[#, z]]& /@ Rest @ CoefficientList[(-Log[1-x]/x)^z + O[x]^(rows+1), x]) * Array[A053657, rows, 2] // Flatten (* Jean-François Alcover, Nov 22 2016 *)

{T(n,k)=local(X=x+x^2*O(x^n)); D=1;for(j=0,n,D=lcm(D,denominator( polcoeff(polcoeff((-log(1-X)/x)^z+z*O(z^j),j,z),n,x)))); return(D*polcoeff(polcoeff((-log(1-X)/x)^z+z*O(z^k),k,z),n,x))}

The n-th row polynomials, P(n, z), satisfy 1 + Sum_{n>=1} P(n, z) x^n = (Sum_{k>=1} x^(k-1)/k)^z.

A163943 Third left hand column of triangle A163940.

Original entry on oeis.org

0, 3, 17, 52, 121, 240, 428, 707, 1102, 1641, 2355, 3278, 4447, 5902, 7686, 9845, 12428, 15487, 19077, 23256, 28085, 33628, 39952, 47127, 55226, 64325, 74503, 85842, 98427, 112346, 127690, 144553, 163032, 183227, 205241, 229180, 255153

Offset: 0

Johannes W. Meijer, Aug 13 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A163972.
Equals the third left hand column of A163940.
A163944 is another left hand column.

LinearRecurrence[{5,-10,10,-5,1},{0,3,17,52,121},40] (* Harvey P. Dale, Feb 25 2017 *)

x='x+O('x^50); concat([0], Vec(x*(3 +2*x -3*x^2 +x^3)/(1-x)^5)) \\ G. C. Greubel, Aug 13 2017

G.f.: x*(3 + 2*x - 3*x^2 + x^3)/(1-x)^5.
a(n)= (2*n + 45*n^2 + 22*n^3 + 3*n^4)/24.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
E.g.f.: (1/24)*x*(72 + 132*x + 40*x^2 + 3*x^3)*exp(x). - G. C. Greubel, Aug 13 2017

A163944 Fourth left hand column of triangle A163940.

Original entry on oeis.org

0, 4, 49, 246, 834, 2250, 5214, 10829, 20696, 37044, 62875, 102124, 159834, 242346, 357504, 514875, 725984, 1004564, 1366821, 1831714, 2421250, 3160794, 4079394, 5210121, 6590424, 8262500, 10273679, 12676824, 15530746, 18900634, 22858500

Offset: 0

Johannes W. Meijer, Aug 13 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A163972.
Equals the fourth left hand column of A163940.
A163943 is another left hand column.

CoefficientList[Series[x*(4 + 21*x - 13*x^2 + x^3 + 3*x^4 - x^5)/(1 - x)^7, {x, 0, 50}], x] (* G. C. Greubel, Aug 13 2017 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,4,49,246,834,2250,5214},40] (* Harvey P. Dale, Apr 29 2019 *)

x='x+O('x^50); concat([0], Vec(x*(4 +21*x -13*x^2 +x^3 +3*x^4 -x^5)/(1-x)^7)) \\ G. C. Greubel, Aug 13 2017

G.f.: x*(4 +21*x -13*x^2 +x^3 +3*x^4 -x^5)/(1-x)^7.
a(n) = (10*n^2 +107*n^3 +61*n^4 +13*n^5 +n^6)/48.
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7).
E.g.f.: (1/48)*x*(192 + 984*x + 888*x^2 + 256*x^3 + 28*x^4 + x^5)*exp(x). - G. C. Greubel, Aug 13 2017

A341111 T(n, k) = [x^k] M(n)Sum_{k=0..n} E2(n, k)binomial(-x + n - k, 2n), where E2 are the second-order Eulerian numbers A340556 and M(n) are the Minkowski numbers A053657. Triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= 2n+1.

Original entry on oeis.org

1, 0, 1, 1, 0, 10, 21, 14, 3, 0, 36, 96, 97, 47, 11, 1, 0, 12048, 36740, 45420, 29855, 11352, 2510, 300, 15, 0, 91200, 304480, 427348, 334620, 162255, 50787, 10302, 1310, 95, 3, 0, 109941120, 392583744, 603023624, 531477324, 300731214, 115291701, 30675678, 5682033, 719866, 59535, 2898, 63

Offset: 0

Peter Luschny, Feb 05 2021

			Triangle starts:
[0] 1;
[1] 0, 1,     1;
[2] 0, 10,    21,     14,     3;
[3] 0, 36,    96,     97,     47,     11,     1;
[4] 0, 12048, 36740,  45420,  29855,  11352,  2510,  300,   15;
[5] 0, 91200, 304480, 427348, 334620, 162255, 50787, 10302, 1310, 95, 3.

Cf. A053657, A163972, A008517, A201637, A340556, A341110 (row sums), A340556.

E2 := (n, k) -> `if`(k=0, k^n, combinat:-eulerian2(n, k-1)):
CoeffList := p -> [op(PolynomialTools:-CoefficientList(p, x))]:
mser := series((y/(exp(y)-1))^x, y, 29): m := n -> denom(coeff(mser, y, n)):
poly := n -> expand(m(n)*add(E2(n, k)*binomial(-x+n-k, 2*n), k = 0..n)):
for n from 0 to 6 do CoeffList(poly(n)) od;

M(n) = prod(i=1, #factor(n!)~, prime(i)^sum(k=0, #binary(n), floor((n-1)/((prime(i)-1)*prime(i)^k)))) \\ from A053657
rows_upto(n) = my(v1, v2); v1 = vector(n, i, 0); v2 = vector(n+1, i, 0); v2[1] = 1; for(i=1, n, v1[i] = (i+x)*(i+x-1)/2*v2[i]; for(j=1, i-1, v1[j] *= (i-j)*(i+x)/(i-j+2)); v2[i+1] = vecsum(v1)/i); v2 = vector(n+1, i, M(i)*Vecrev(v2[i])) \\ Mikhail Kurkov, Aug 27 2025

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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A163940 Triangle related to the divergent series 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ... for m >= -1.

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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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A075264 Triangle of numerators of coefficients, where the n-th row forms the polynomial in z, P(n,z), that is the coefficient of x^n in {-log(1-x)/x}^z, for n > 0. The denominator for all the terms in the n-th row is A053657(n).

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A163943 Third left hand column of triangle A163940.

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A163944 Fourth left hand column of triangle A163940.

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A341111 T(n, k) = [x^k] M(n)*Sum_{k=0..n} E2(n, k)*binomial(-x + n - k, 2*n), where E2 are the second-order Eulerian numbers A340556 and M(n) are the Minkowski numbers A053657. Triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= 2*n+1.

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A163940 Triangle related to the divergent series 1^m1! - 2^m2! + 3^m3! - 4^m4! + ... for m >= -1.

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).

A341111 T(n, k) = [x^k] M(n)Sum_{k=0..n} E2(n, k)binomial(-x + n - k, 2n), where E2 are the second-order Eulerian numbers A340556 and M(n) are the Minkowski numbers A053657. Triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= 2n+1.