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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"7c9e9b264737ebcc1bafefba52921f5ca17ace0bd017394b92b0dfc257af1a07","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-0e6d847331141ca6","role":"sequence","name":"basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1","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-0e6d847331141ca6","role":"sequence","name":"basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1","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:  3,   2;\n  1:  5,  10,    3;\n  2:  7,  28,   23,     4;\n  3:  9,  60,   95,    44,      5;\n  4: 11, 110,  285,   250,     75,      6;\n  5: 13, 182,  700,   994,    560,    118,     7;\n  6: 15, 280, 1498,  3136,   2842,   1120,   175,     8;\n  7: 17, 408, 2898,  8400,  11172,   7056,  2058,   248,    9;\n  8: 19, 570, 5190, 19908,  36540,  33768, 15750,  3540,  339,  10;\n  9: 21, 770, 8745, 42900, 103950, 132264, 90090, 32340, 5775, 450, 11;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3+32*n^2+65*n+25+(n^3+7*n^2+11*n+5)*x^2)*P(n,x) = (4*n^3+36*n^2+95*n+75+(12*n^3+64*n^2+102*n+50)*x+(n^3+12*n^2+26*n+15)*x^2+(3*n^3+14*n^2+21*n+10)*x^3)*P(n-1,x) + ((8*n^2+36*n+34)*x-(8*n^2+30*n+22)*x^2+(2*n^2-8)*x^3-(2*n^2+6*n+4)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 3 + 2*x.","mathematica":"P[0, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^3 + 36*n^2 + 95*n + 75 + (12*n^3 + 64*n^2 + 102*n + 50)*x + (n^3 + 12*n^2 + 26*n + 15)*x^2 + (3*n^3 + 14*n^2 + 21*n + 10)*x^3)*P[n - 1, x] + ((8*n^2 + 36*n + 34)*x - (8*n^2 + 30*n + 22)*x^2 + (2*n^2 - 8)*x^3 - (2*n^2 + 6*n + 4)*x^4)*D[P[n - 1, x], x])/(4*n^3 + 32*n^2 + 65*n + 25 + (n^3 + 7*n^2 + 11*n + 5)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^3 + 36*n^2 + 95*n + 75 + (12*n^3 + 64*n^2 + 102*n + 50)*x + (n^3 + 12*n^2 + 26*n + 15)*x^2 + (3*n^3 + 14*n^2 + 21*n + 10)*x^3)*v[i - 1] + ((8*n^2 + 36*n + 34)*x - (8*n^2 + 30*n + 22)*x^2 + (2*n^2 - 8)*x^3 - (2*n^2 + 6*n + 4)*x^4)*deriv(v[i - 1], x))/(4*n^3 + 32*n^2 + 65*n + 25 + (n^3 + 7*n^2 + 11*n + 5)*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-0e6d847331141ca6","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-455a924384122a0c"],"text":"A145596 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A145596 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}