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).

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

0,5

Comments

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

Links

Table of n, a(n) for n=0..63.

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

Formula

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.

Example

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.

Mathematica

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 *)

Crossrefs

Cf. A075264, A053657, A163972.

Sequence in context: A373128 A222345 A353618 * A234642 A038228 A213214

Adjacent sequences: A202336 A202337 A202338 * A202340 A202341 A202342

Keyword

sign,tabf

Author

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

Status

approved