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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":"3154ee4be2d2f4e34f1af9b431dbc8685a524c73fc934421fd573b1b39b105c3","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-f562ab3e3b669a1b","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","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:         11,          26;\n  2:        121,         572,          676;\n  3:       1331,        9438,        22308,         17576;\n  4:      14641,      138424,       490776,        773344,        456976;\n  5:     161051,     1903330,      8997560,      21266960,      25133680,       11881376;\n  6:    1771561,    25123956,    148459740,     467873120,     829411440,      784170816,      308915776;\n  7:   19487171,   322424102,   2286279996,    9006557560,   21288226960,    30190576416,    23786514752,     8031810176;\n  8:  214358881,  4053331568,  33532106608,  158515413056,  468340993120,   885590241536,  1046606649088,   706799295488,   208827064576;\n  9: 2357947691, 50159978154, 474239793456, 2615504315424, 9273151663776, 21918358478016, 34538019419904, 34986565126656, 20673879393024, 5429503678976;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (11+37*x+26*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[((11 + 37*x + 26*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] = ((11 + 37*x + 26*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-e6ebe74518d42cbe","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-aef71e51dba5316b","e-414514ad487c1f5e"],"text":"A038259 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A038259 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}