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b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"7b7b5339664f870d13ae8c64e34cf34adb090111f5a4b480fe1af0be7a10ada1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-623036dc25b33b8c","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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:           2,            4;\n  2:           8,           36,            40;\n  3:          56,          404,           964,            760;\n  4:         616,         6180,         23128,          38244,          23560;\n  5:        9856,       127216,        654328,        1675792,        2136184,         1083760;\n  6:      216832,      3429536,      22537040,       78744416,      154246736,       160558496,        69360640;\n  7:     6288128,    117887264,     945084720,     4199236464,    11166430704,     17767168944,     15658930720,      5895654400;\n  8:   232660736,   5047234720,   47817846416,   258385983648,   870874710624,   1874526197664,   2516001851536,   1924962661280,    642626329600;\n  9: 10702393856, 263814657216, 2886044857056, 18388982360384, 75200730464832, 204667165737408, 370671648052960, 430724534227776, 291355733095680, 87397180825600;\n","recurrence_sentence":"The row polynomials satisfy (2+2*x)*P(n,x) = (n^2+n+2+(4*n^2+4*n+4)*x+(3*n^2+3*n+2)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + 2 + (4*n^2 + 4*n + 4)*x + (3*n^2 + 3*n + 2)*x^2)*P[n - 1, x])/(2 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + n + 2 + (4*n^2 + 4*n + 4)*x + (3*n^2 + 3*n + 2)*x^2)*v[i - 1])/(2 + 2*x)); 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-ecd77b030b8fab94","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-42bb566932c0b2ac","e-653be21178f4fe4e"],"text":"A128813 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A128813 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}