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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":"69fcfbdd1e032231b39f1f53b2719ec83101f16856f749a3a674d0cf62c390cc","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-2c8bb380cc75972b","role":"sequence","name":"Triangle related to the asymptotic expansion of E(x,m=3,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,         6;\n  2:       11,        40,        35;\n  3:       50,       255,       420,        225;\n  4:      274,      1771,      4179,       4291,       1624;\n  5:     1764,     13692,     41538,      61852,      45353,      13132;\n  6:    13068,    117804,    433614,     837376,     897294,     506892,     118124;\n  7:   109584,   1121460,   4831200,   11394030,   15925920,   13216410,    6037860,    1172700;\n  8:  1026576,  11730708,  57712908,  160114086,  274540365,  298390686,  201002208,   76800108,   12753576;\n  9: 10628640, 133918488, 739173864, 2351474444, 4759953033, 6366961458, 5633856910, 3182827692, 1042487556, 150917976;\n","recurrence_sentence":"The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (n^3-3*n^2+3*n-1+(6*n^3-15*n^2+12*n-3)*x+(15*n^3-30*n^2+18*n-3)*x^2+(20*n^3-30*n^2+12*n-1)*x^3+(15*n^3-15*n^2+3*n)*x^4+(6*n^3-3*n^2)*x^5+n^3*x^6)*P(n-3,x) - (3*n^2-3*n+1+(15*n^2-9*n+2)*x+(30*n^2-6*n+1)*x^2+(30*n^2+6*n+1)*x^3+(15*n^2+9*n+2)*x^4+(3*n^2+3*n+1)*x^5)*P(n-2,x) + (3*n+(12*n+3)*x+(18*n+9)*x^2+(12*n+9)*x^3+(3*n+3)*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 6*x, P(2,x) = 11 + 40*x + 35*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 6*x;\nP[2, x_] := 11 + 40*x + 35*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 - 3*n^2 + 3*n - 1 + (6*n^3 - 15*n^2 + 12*n - 3)*x + (15*n^3 - 30*n^2 + 18*n - 3)*x^2 + (20*n^3 - 30*n^2 + 12*n - 1)*x^3 + (15*n^3 - 15*n^2 + 3*n)*x^4 + (6*n^3 - 3*n^2)*x^5 + n^3*x^6)*P[n - 3, x] - (3*n^2 - 3*n + 1 + (15*n^2 - 9*n + 2)*x + (30*n^2 - 6*n + 1)*x^2 + (30*n^2 + 6*n + 1)*x^3 + (15*n^2 + 9*n + 2)*x^4 + (3*n^2 + 3*n + 1)*x^5)*P[n - 2, x] + (3*n + (12*n + 3)*x + (18*n + 9)*x^2 + (12*n + 9)*x^3 + (3*n + 3)*x^4)*P[n - 1, x])/(1 + 3*x + 3*x^2 + x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 6*x; v[3] = 11 + 40*x + 35*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 - 3*n^2 + 3*n - 1 + (6*n^3 - 15*n^2 + 12*n - 3)*x + (15*n^3 - 30*n^2 + 18*n - 3)*x^2 + (20*n^3 - 30*n^2 + 12*n - 1)*x^3 + (15*n^3 - 15*n^2 + 3*n)*x^4 + (6*n^3 - 3*n^2)*x^5 + n^3*x^6)*v[i - 3] - (3*n^2 - 3*n + 1 + (15*n^2 - 9*n + 2)*x + (30*n^2 - 6*n + 1)*x^2 + (30*n^2 + 6*n + 1)*x^3 + (15*n^2 + 9*n + 2)*x^4 + (3*n^2 + 3*n + 1)*x^5)*v[i - 2] + (3*n + (12*n + 3)*x + (18*n + 9)*x^2 + (12*n + 9)*x^3 + (3*n + 3)*x^4)*v[i - 1])/(1 + 3*x + 3*x^2 + 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-efe23a76953cf69d","steps":["D\u00b2\u2218x (shift +1)","x \u21a6 x/(x+1)"],"edges":["e-337faa044bd16e1a","e-9ea310aace5b454f"],"text":"A130534 \u2192 D\u00b2\u2218x (shift +1) \u2192 x \u21a6 x/(x+1)"},"headline":"A130534 \u2192 D\u00b2\u2218x (shift +1) \u2192 x \u21a6 x/(x+1)","chain":null}