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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"100d5cf26167c20550d198026af82ab78b2aee2436e5c0f64d465dd0f7c7c460","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-97228a88c81561dc","role":"sequence","name":"Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","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-97228a88c81561dc","role":"sequence","name":"Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","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:    7,     6;\n  1:   27,    48,     21;\n  2:   77,   210,    189,     56;\n  3:  182,   672,    924,    560,    126;\n  4:  378,  1764,   3276,   3024,   1386,     252;\n  5:  714,  4032,   9450,  11760,   8190,    3024,    462;\n  6: 1254,  8316,  23562,  36960,  34650,   19404,   6006,    792;\n  7: 2079, 15840,  52668,  99792, 117810,   88704,  41580,  11088,   1287;\n  8: 3289, 28314, 108108, 240240, 342342,  324324, 204204,  82368,  19305,  2002;\n  9: 5005, 48048, 207207, 528528, 882882, 1009008, 798798, 432432, 153153, 32032, 3003;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+7*n+5+(n^2+6*n+5)*x)*P(n,x) = (2*n^2+17*n+35+(3*n^2+28*n+65)*x+(n^2+11*n+30)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 7 + 6*x.","mathematica":"P[0, x_] := 7 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 17*n + 35 + (3*n^2 + 28*n + 65)*x + (n^2 + 11*n + 30)*x^2)*P[n - 1, x])/(2*n^2 + 7*n + 5 + (n^2 + 6*n + 5)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 7 + 6*x; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + 17*n + 35 + (3*n^2 + 28*n + 65)*x + (n^2 + 11*n + 30)*x^2)*v[i - 1])/(2*n^2 + 7*n + 5 + (n^2 + 6*n + 5)*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-97228a88c81561dc","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-c021bdf930303dcb"],"text":"A178822 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A178822 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}