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. Triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= 2*n+1.

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

0,6

Comments

The Minkowski numbers (Minkowski, 1887, p.201) are the denominators of the y-coefficients of the series (y/(exp(y) - 1))^x.

Links

Table of n, a(n) for n=0..48.

Hermann Minkowski, Zur Theorie der quadratischen Formen, J. Reine Angew. Math. 101 (1887), 196-202. ( = Ges. Abh., pp. 212-218, Chelsea, New York, 1967.)

Example

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;
  ...

Maple

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;

Prog

(PARI) 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(n) = my(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

Crossrefs

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

Sequence in context: A160479 A085222 A085221 * A397212 A128536 A202318

Adjacent sequences: A341108 A341109 A341110 * A341112 A341113 A341114

Keyword

nonn,tabf

Author

Peter Luschny, Feb 05 2021

Status

approved