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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"28ddaddb1ea2be9a20359679e1143ad6fbe297eb9c15c72ffe14495b931e66a6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-4ca874876022e136","role":"sequence","name":"Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-4ca874876022e136","role":"sequence","name":"Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"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: 2,   3;\n  1: 2,  11,    6;\n  2: 2,  23,   36,    10;\n  3: 2,  39,  120,    90,     15;\n  4: 2,  59,  300,   430,    190,     21;\n  5: 2,  83,  630,  1470,   1225,    357,     28;\n  6: 2, 111, 1176,  4060,   5460,   2982,    616,     36;\n  7: 2, 143, 2016,  9660,  19110,  16758,   6468,    996,    45;\n  8: 2, 179, 3240, 20580,  56280,  72324,  44688,  12840,  1530,   55;\n  9: 2, 219, 4950, 40260, 145530, 257796, 232848, 106920, 23760, 2255, 66;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4+24*n^3+44*n^2+24*n-(2*n^3+6*n^2-2*n-6)*x+(n^4+6*n^3+11*n^2+6*n)*x^2)*P(n,x) = (4*n^4+24*n^3+44*n^2+24*n+(12*n^4+58*n^3+92*n^2+60*n+12)*x+(n^4+4*n^3+13*n^2+24*n+18)*x^2+(3*n^4+16*n^3+29*n^2+18*n)*x^3)*P(n-1,x) + ((8*n^3+36*n^2+44*n+12)*x-(8*n^3+34*n^2+30*n-6)*x^2+(2*n^3+6*n^2-6*n-18)*x^3-(2*n^3+8*n^2+8*n)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + 3*x.","mathematica":"P[0, x_] := 2 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^4 + 24*n^3 + 44*n^2 + 24*n + (12*n^4 + 58*n^3 + 92*n^2 + 60*n + 12)*x + (n^4 + 4*n^3 + 13*n^2 + 24*n + 18)*x^2 + (3*n^4 + 16*n^3 + 29*n^2 + 18*n)*x^3)*P[n - 1, x] + ((8*n^3 + 36*n^2 + 44*n + 12)*x - (8*n^3 + 34*n^2 + 30*n - 6)*x^2 + (2*n^3 + 6*n^2 - 6*n - 18)*x^3 - (2*n^3 + 8*n^2 + 8*n)*x^4)*D[P[n - 1, x], x])/(4*n^4 + 24*n^3 + 44*n^2 + 24*n - (2*n^3 + 6*n^2 - 2*n - 6)*x + (n^4 + 6*n^3 + 11*n^2 + 6*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + 3*x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^4 + 24*n^3 + 44*n^2 + 24*n + (12*n^4 + 58*n^3 + 92*n^2 + 60*n + 12)*x + (n^4 + 4*n^3 + 13*n^2 + 24*n + 18)*x^2 + (3*n^4 + 16*n^3 + 29*n^2 + 18*n)*x^3)*v[i - 1] + ((8*n^3 + 36*n^2 + 44*n + 12)*x - (8*n^3 + 34*n^2 + 30*n - 6)*x^2 + (2*n^3 + 6*n^2 - 6*n - 18)*x^3 - (2*n^3 + 8*n^2 + 8*n)*x^4)*deriv(v[i - 1], x))/(4*n^4 + 24*n^3 + 44*n^2 + 24*n - (2*n^3 + 6*n^2 - 2*n - 6)*x + (n^4 + 6*n^3 + 11*n^2 + 6*n)*x^2)); 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-4ca874876022e136","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-039745f356dca7e2"],"text":"A062196 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A062196 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}