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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":"69f8717d177ae721999dc4927cb54e26bba969015d2a084a3fdbc86cfee7bab8","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-1397bdceeefc7b3d","role":"sequence","name":"Triangle read by rows T(n, k) = binomial(2*n, k) * binomial(3*n - k, 2*n).","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,        5;\n  2:      15,       50,        41;\n  3:      84,      420,       693,        377;\n  4:     495,     3300,      8190,       8964,       3649;\n  5:    3003,    25025,     82940,     136620,     111815,      36365;\n  6:   18564,   185640,    769860,    1694420,    2087085,    1363830,     369305;\n  7:  116280,  1356600,   6755868,   18613980,   30640120,   30128280,   16383465,    3800305;\n  8:  735471,  9806280,  57004332,  188675928,  388869390,  511004088,  418054140,  194654216,   39490049;\n  9: 4686825, 70302375, 467240400, 1805739936, 4471750206, 7358114610, 8044270920, 5633882280, 2293485003, 413442773;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-2*n+(12*n^2-6*n)*x+(12*n^2-6*n)*x^2+(4*n^2-2*n)*x^3)*P(n,x) = (27*n^2-27*n+6+(141*n^2-146*n+33)*x+(281*n^2-302*n+69)*x^2+(267*n^2-300*n+69)*x^3+(120*n^2-143*n+33)*x^4+(20*n^2-26*n+6)*x^5)*P(n-1,x) - ((9*n-3)*x+(55*n-18)*x^2+(131*n-42)*x^3+(153*n-48)*x^4+(88*n-27)*x^5+(20*n-6)*x^6)*d/dx 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[((27*n^2 - 27*n + 6 + (141*n^2 - 146*n + 33)*x + (281*n^2 - 302*n + 69)*x^2 + (267*n^2 - 300*n + 69)*x^3 + (120*n^2 - 143*n + 33)*x^4 + (20*n^2 - 26*n + 6)*x^5)*P[n - 1, x] - ((9*n - 3)*x + (55*n - 18)*x^2 + (131*n - 42)*x^3 + (153*n - 48)*x^4 + (88*n - 27)*x^5 + (20*n - 6)*x^6)*D[P[n - 1, x], x])/(4*n^2 - 2*n + (12*n^2 - 6*n)*x + (12*n^2 - 6*n)*x^2 + (4*n^2 - 2*n)*x^3)]];\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] = ((27*n^2 - 27*n + 6 + (141*n^2 - 146*n + 33)*x + (281*n^2 - 302*n + 69)*x^2 + (267*n^2 - 300*n + 69)*x^3 + (120*n^2 - 143*n + 33)*x^4 + (20*n^2 - 26*n + 6)*x^5)*v[i - 1] - ((9*n - 3)*x + (55*n - 18)*x^2 + (131*n - 42)*x^3 + (153*n - 48)*x^4 + (88*n - 27)*x^5 + (20*n - 6)*x^6)*deriv(v[i - 1], x))/(4*n^2 - 2*n + (12*n^2 - 6*n)*x + (12*n^2 - 6*n)*x^2 + (4*n^2 - 2*n)*x^3)); 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-1397bdceeefc7b3d","steps":["x \u21a6 x/(x+1)"],"edges":["e-add577d5584f6659"],"text":"A357613 \u2192 x \u21a6 x/(x+1)"},"headline":"A357613 \u2192 x \u21a6 x/(x+1)","chain":null}