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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). 8

%I

%S 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,

%T 3,55,410,1598,3467,4055,2120,52,-240,0,0,63,1638,17955,107954,387009,

%U 837426,1038681,606606,9828,-113624,-2016,11520,0,9,315,4767,40859,217973,747021,1628877,2122953,1344798,-5516,-374024,-2592,80640,0,0

%N 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).

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

%H Norman Do and Paul Norbury, <a href="http://arxiv.org/abs/1312.7516">Pruned Hurwitz numbers</a>, arXiv preprint arXiv:1312.7516 [math.GT], 2013.

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

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

%e Q_0 = 1,

%e Q_1 = (x^2 + x)/2,

%e Q_2 = (3x^4 + 10x^3 + 9x^2 + 2x)/24,

%e Q_3 = (x^6 + 7x^5 + 17x^4 + 17x^3 + 6x^2)/48,

%e Q_4 = (15x^8 + 180x^7 + 830x^6 + 1848x^5 + 2015x^4 + 900x^3 + 20x^2 -48x)/5760,

%e Q_5 = (3x^10 + 55x^9 + 410x^8 + 1598x^7 + 3467x^6 + 4055x^5 + 2120x^4 + 52x^3 -240x^2)/11520,

%e 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,

%e 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,

%e 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.

%t 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-Fran├žois Alcover_, Nov 22 2016 *)

%Y Cf. A075264, A053657, A163972.

%K sign,tabf

%O 0,5

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Dec 17 2011

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Last modified June 18 10:54 EDT 2021. Contains 345098 sequences. (Running on oeis4.)