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for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"0e08866b2c1f319f337e5c0e0b23bc8240b762879f68de18df5e82938c5efa9c","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-70594f2875298da0","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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:        3,         9;\n  2:       12,        75,        117;\n  3:       60,       579,       1860,        1989;\n  4:      360,      4734,      23319,       50994,        41769;\n  5:     2520,     42138,     281583,      939933,      1567233,       1044225;\n  6:    20160,    410184,    3474666,    15685371,     39795921,      53803557,      30282525;\n  7:   181440,   4356936,   44808066,   255832317,    875780532,    1797497406,    2048060106,     999323325;\n  8:  1814400,  50282640,  609287292,  4216221612,  18223601049,   50378853744,   86988005082,   85771457172,   36974963025;\n  9: 19958400, 627499440, 8763748452, 71359216704, 373324697631, 1301335034193, 3022401059406, 4509994237254, 3923354337327, 1515973484025;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n+2+(5*n+7)*x+(4*n+5)*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 + (5*n + 7)*x + (4*n + 5)*x^2)*P[n - 1, x])/(1 + 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 + (5*n + 7)*x + (4*n + 5)*x^2)*v[i - 1])/(1 + 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-2feb6dc01f424b0a","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-45b4054949a52789","e-c159b21cf2bab04e"],"text":"A145324 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A145324 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}