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decompose into exactly k cycles of even length, k=0..n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:                1;\n  1:                4,                 3;\n  2:               45,                90,                27;\n  3:             1260,              3465,              2430,               405;\n  4:            66465,            217980,            217350,             79380,              8505;\n  5:          5681340,          20947815,          25685100,          13522950,           3061800,            229635;\n  6:        716911965,        2880184230,        4065806745,        2677544100,         879884775,         136403190,          7577955;\n  7:     125372307060,      538456083165,      843739426530,      649163269755,      269707838400,       60914128275,       6895939050,       295540245;\n  8:   29030535158625,   131614360277400,   223777162709100,   193451011954200,    94844447047350,    27280334881800,    4509944138700,    390113123400,    13299311025;\n  9: 8599375008724500, 40788114358466475, 74104595111547000, 70217301947492700, 39018216160123200, 13340063069383050, 2823390533363400, 357219494131500, 24417535041900, 678264862275;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(48*n^3-168*n^2+180*n-54)*x+(36*n^2-72*n+27)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(12*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (48*n^3 - 168*n^2 + 180*n - 54)*x + (36*n^2 - 72*n + 27)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (12*n - 6)*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (48*n^3 - 168*n^2 + 180*n - 54)*x + (36*n^2 - 72*n + 27)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (12*n - 6)*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-1336d058dca9979e","steps":["basis A013610","D+1"],"edges":["e-05481f5802c9cdfa","e-d5a5e4f6d2369c49"],"text":"A204420 \u2192 basis A013610 \u2192 D+1"},"headline":"A204420 \u2192 basis A013610 \u2192 D+1","chain":null}