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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":"99de43927c3edb4c779c3f4643a7393ba5355062cd9829f4d74c7513d8c2b4f9","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-2f0c73de4c291a40","role":"sequence","name":"basis_map_A013611 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,        11;\n  2:       12,        92,         176;\n  3:       60,       712,        2812,         3696;\n  4:      360,      5832,       35384,        95288,        96096;\n  5:     2520,     51984,      428480,      1763920,      3626600,       2978976;\n  6:    20160,    506592,     5299264,     29536640,     92513920,     154389408,      107243136;\n  7:   181440,   5385888,    68463648,    483099584,   2043627520,    5182575392,     7295153952,     4396968576;\n  8:  1814400,  62205120,   932387328,   7980323648,  42658856064,  145832619840,   311350007552,   379546767552,   202260554496;\n  9: 19958400, 776790720, 13428721728, 135335313856, 876243922752, 3779760477504, 10862313694912, 20053864828224, 21581751244608, 10315288279296;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n+2+(6*n+8)*x+(5*n+6)*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 + (6*n + 8)*x + (5*n + 6)*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 + (6*n + 8)*x + (5*n + 6)*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 A013611","x \u21a6 x/(x+1)"],"edges":["e-39cad9e518a532bb","e-98cd98f3cc2359bf"],"text":"A145324 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A145324 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}