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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"b49f3e107af112de5d40b608b1d820d4535eb12923403b7a7f4fda674dc90abc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ef0a5df5e949a196","role":"sequence","name":"Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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:      2,        1;\n  2:     12,       13,          1;\n  3:     55,      162,         57,          1;\n  4:    230,     1807,       1476,        233,          1;\n  5:    933,    16882,      31900,      11703,        937,          1;\n  6:   3748,   141003,     554229,     436379,      87363,       3753,         1;\n  7:  15011,  1101298,    8147421,   12673795,    5250479,     633204,     15017,        1;\n  8:  60066,  8256943,  106829773,  299801118,  237608225,   58834412,   4518894,    60073,      1;\n  9: 240289, 60351354, 1295868192, 6096201873, 8511396480, 3950191398, 632326284, 31977837, 240297, 1;\n","recurrence_sentence":"The row polynomials satisfy 8*P(n,x) = (54*n^2-162*n+96+(27*n^2-81*n-60)*x-126*x^2-(81*n^2-324*n+252)*x^3)*P(n-4,x) + ((27*n^2-81*n+48)*x-54*x^2-(81*n^2-324*n+252)*x^3)*d/dx P(n-4,x) + (54*n^3-351*n^2+663*n-336+(27*n^3-99*n^2-114*n+586)*x-(27*n^2+90*n-480)*x^2-(81*n^3-540*n^2+1116*n-528)*x^3)*P(n-3,x) + (63*n^2-189*n+112+(27*n^3-234*n^2+507*n-398)*x-(54*n^2-135*n-174)*x^2-(81*n^3-621*n^2+1440*n-978)*x^3+(81*n^2-324*n+252)*x^4)*d/dx P(n-3,x) + ((27*n^2-81*n+48)*x-(27*n^2-81*n+102)*x^2-(81*n^2-324*n+198)*x^3+(81*n^2-324*n+252)*x^4)*d^2/dx^2 P(n-3,x) - (9*n^2+45*n-48+(108*n^2-582*n+454)*x+(45*n^2-240*n+344)*x^2+(144*n-240)*x^3)*P(n-2,x) - (9*n^2-27*n+16+(54*n-234)*x-(27*n^2-54*n-240)*x^2+36*x^3-144*x^4)*d/dx P(n-2,x) + (72*x^2-144*x^3+72*x^4)*d^2/dx^2 P(n-2,x) + (18*n+16+(48*n-164)*x+48*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 12 + 13*x + x^2, P(3,x) = 55 + 162*x + 57*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 12 + 13*x + x^2;\nP[3, x_] := 55 + 162*x + 57*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((54*n^2 - 162*n + 96 + (27*n^2 - 81*n - 60)*x - 126*x^2 - (81*n^2 - 324*n + 252)*x^3)*P[n - 4, x] + ((27*n^2 - 81*n + 48)*x - 54*x^2 - (81*n^2 - 324*n + 252)*x^3)*D[P[n - 4, x], x] + (54*n^3 - 351*n^2 + 663*n - 336 + (27*n^3 - 99*n^2 - 114*n + 586)*x - (27*n^2 + 90*n - 480)*x^2 - (81*n^3 - 540*n^2 + 1116*n - 528)*x^3)*P[n - 3, x] + (63*n^2 - 189*n + 112 + (27*n^3 - 234*n^2 + 507*n - 398)*x - (54*n^2 - 135*n - 174)*x^2 - (81*n^3 - 621*n^2 + 1440*n - 978)*x^3 + (81*n^2 - 324*n + 252)*x^4)*D[P[n - 3, x], x] + ((27*n^2 - 81*n + 48)*x - (27*n^2 - 81*n + 102)*x^2 - (81*n^2 - 324*n + 198)*x^3 + (81*n^2 - 324*n + 252)*x^4)*D[P[n - 3, x], {x, 2}] - (9*n^2 + 45*n - 48 + (108*n^2 - 582*n + 454)*x + (45*n^2 - 240*n + 344)*x^2 + (144*n - 240)*x^3)*P[n - 2, x] - (9*n^2 - 27*n + 16 + (54*n - 234)*x - (27*n^2 - 54*n - 240)*x^2 + 36*x^3 - 144*x^4)*D[P[n - 2, x], x] + (72*x^2 - 144*x^3 + 72*x^4)*D[P[n - 2, x], {x, 2}] + (18*n + 16 + (48*n - 164)*x + 48*x^2)*P[n - 1, x])/(8)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 12 + 13*x + x^2; v[4] = 55 + 162*x + 57*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((54*n^2 - 162*n + 96 + (27*n^2 - 81*n - 60)*x - 126*x^2 - (81*n^2 - 324*n + 252)*x^3)*v[i - 4] + ((27*n^2 - 81*n + 48)*x - 54*x^2 - (81*n^2 - 324*n + 252)*x^3)*deriv(v[i - 4], x) + (54*n^3 - 351*n^2 + 663*n - 336 + (27*n^3 - 99*n^2 - 114*n + 586)*x - (27*n^2 + 90*n - 480)*x^2 - (81*n^3 - 540*n^2 + 1116*n - 528)*x^3)*v[i - 3] + (63*n^2 - 189*n + 112 + (27*n^3 - 234*n^2 + 507*n - 398)*x - (54*n^2 - 135*n - 174)*x^2 - (81*n^3 - 621*n^2 + 1440*n - 978)*x^3 + (81*n^2 - 324*n + 252)*x^4)*deriv(v[i - 3], x) + ((27*n^2 - 81*n + 48)*x - (27*n^2 - 81*n + 102)*x^2 - (81*n^2 - 324*n + 198)*x^3 + (81*n^2 - 324*n + 252)*x^4)*deriv(deriv(v[i - 3], x), x) - (9*n^2 + 45*n - 48 + (108*n^2 - 582*n + 454)*x + (45*n^2 - 240*n + 344)*x^2 + (144*n - 240)*x^3)*v[i - 2] - (9*n^2 - 27*n + 16 + (54*n - 234)*x - (27*n^2 - 54*n - 240)*x^2 + 36*x^3 - 144*x^4)*deriv(v[i - 2], x) + (72*x^2 - 144*x^3 + 72*x^4)*deriv(deriv(v[i - 2], x), x) + (18*n + 16 + (48*n - 164)*x + 48*x^2)*v[i - 1])/(8)); 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-ef0a5df5e949a196","steps":["D+1"],"edges":["e-4d39029882ca9b9a"],"text":"A144440 \u2192 D+1"},"headline":"A144440 \u2192 D+1","chain":null}