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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":"76442088605fd4ef821956157a538b0350f816952c85ca72d68b20bcfe1e1439","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-7ce10652a7d44cf0","role":"sequence","name":"basis_map_A013609 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,         7;\n  2:       12,        58,         70;\n  3:       60,       446,       1104,         910;\n  4:      360,      3636,      13760,       23124,        14560;\n  5:     2520,     32292,     165404,      423308,       541276,       276640;\n  6:    20160,    313776,    2033656,     7025352,     13642984,     14121192,      6086080;\n  7:   181440,   3327984,   26147304,   114069568,    298420656,    468165328,    407804520,    152152000;\n  8:  1814400,  38360160,  354656592,  1872820192,   6178154464,  13037431648,  17186674384,  12940046560,   4260256000;\n  9: 19958400, 478208160, 5090387472, 31595376464, 126017125056, 334934536512, 593213799312, 675127418064, 448004259360, 132067936000;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n+2+(4*n+6)*x+(3*n+4)*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 + (4*n + 6)*x + (3*n + 4)*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 + (4*n + 6)*x + (3*n + 4)*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 A013609","x \u21a6 x/(x+1)"],"edges":["e-5f6e941a4a66ef73","e-f6ccaa4f7b23b81d"],"text":"A145324 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A145324 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}