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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":"3d8a0b5a198167b11b13608d64e0b0b90f667f3a3e7f3e774b74f4760691f595","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-723ec2078be12613","role":"sequence","name":"Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*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:      15,        16,          1;\n  3:      84,       249,         86,           1;\n  4:     433,      3532,       2846,         436,           1;\n  5:    2182,     41977,      79592,       28622,        2186,           1;\n  6:   10931,    446704,    1788531,     1395096,      272537,       10936,          1;\n  7:   54680,   4453153,   33968622,    52948975,    21585980,     2527311,      54686,         1;\n  8:  273429,  42681084,  575372408,  1633311748,  1285761522,   311945828,   23117824,    273436,       1;\n  9: 1367178, 399332337, 9019852064, 43252457884, 60468252084, 27719780982, 4333167888, 209919996, 1367186, 1;\n","recurrence_sentence":"The row polynomials satisfy 32*P(n,x) = (384*n^2-1216*n+824+(192*n^2-608*n-356)*x-896*x^2-(768*n^2-3200*n+2736)*x^3)*P(n-4,x) + ((192*n^2-608*n+412)*x-384*x^2-(768*n^2-3200*n+2736)*x^3)*d/dx P(n-4,x) + (384*n^3-2608*n^2+5232*n-2987+(192*n^3-752*n^2-668*n+4331)*x-(192*n^2+608*n-3444)*x^2-(768*n^3-5312*n^2+11536*n-6500)*x^3)*P(n-3,x) + (432*n^2-1368*n+927+(192*n^3-1712*n^2+3908*n-3233)*x-(384*n^2-1024*n-1112)*x^2-(768*n^3-6080*n^2+14736*n-10644)*x^3+(768*n^2-3200*n+2736)*x^4)*d/dx P(n-3,x) + ((192*n^2-608*n+412)*x-(192*n^2-608*n+796)*x^2-(768*n^2-3200*n+2352)*x^3+(768*n^2-3200*n+2736)*x^4)*d^2/dx^2 P(n-3,x) - (48*n^2+328*n-417+(768*n^2-4352*n+4080)*x+(320*n^2-1760*n+2484)*x^2+(1024*n-1792)*x^3)*P(n-2,x) - (48*n^2-152*n+103+(384*n-1664)*x-(192*n^2-416*n-1652)*x^2+256*x^3-1024*x^4)*d/dx P(n-2,x) + (512*x^2-1024*x^3+512*x^4)*d^2/dx^2 P(n-2,x) + (96*n+56+(256*n-896)*x+256*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 15 + 16*x + x^2, P(3,x) = 84 + 249*x + 86*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 15 + 16*x + x^2;\nP[3, x_] := 84 + 249*x + 86*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((384*n^2 - 1216*n + 824 + (192*n^2 - 608*n - 356)*x - 896*x^2 - (768*n^2 - 3200*n + 2736)*x^3)*P[n - 4, x] + ((192*n^2 - 608*n + 412)*x - 384*x^2 - (768*n^2 - 3200*n + 2736)*x^3)*D[P[n - 4, x], x] + (384*n^3 - 2608*n^2 + 5232*n - 2987 + (192*n^3 - 752*n^2 - 668*n + 4331)*x - (192*n^2 + 608*n - 3444)*x^2 - (768*n^3 - 5312*n^2 + 11536*n - 6500)*x^3)*P[n - 3, x] + (432*n^2 - 1368*n + 927 + (192*n^3 - 1712*n^2 + 3908*n - 3233)*x - (384*n^2 - 1024*n - 1112)*x^2 - (768*n^3 - 6080*n^2 + 14736*n - 10644)*x^3 + (768*n^2 - 3200*n + 2736)*x^4)*D[P[n - 3, x], x] + ((192*n^2 - 608*n + 412)*x - (192*n^2 - 608*n + 796)*x^2 - (768*n^2 - 3200*n + 2352)*x^3 + (768*n^2 - 3200*n + 2736)*x^4)*D[P[n - 3, x], {x, 2}] - (48*n^2 + 328*n - 417 + (768*n^2 - 4352*n + 4080)*x + (320*n^2 - 1760*n + 2484)*x^2 + (1024*n - 1792)*x^3)*P[n - 2, x] - (48*n^2 - 152*n + 103 + (384*n - 1664)*x - (192*n^2 - 416*n - 1652)*x^2 + 256*x^3 - 1024*x^4)*D[P[n - 2, x], x] + (512*x^2 - 1024*x^3 + 512*x^4)*D[P[n - 2, x], {x, 2}] + (96*n + 56 + (256*n - 896)*x + 256*x^2)*P[n - 1, x])/(32)]];\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] = 15 + 16*x + x^2; v[4] = 84 + 249*x + 86*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((384*n^2 - 1216*n + 824 + (192*n^2 - 608*n - 356)*x - 896*x^2 - (768*n^2 - 3200*n + 2736)*x^3)*v[i - 4] + ((192*n^2 - 608*n + 412)*x - 384*x^2 - (768*n^2 - 3200*n + 2736)*x^3)*deriv(v[i - 4], x) + (384*n^3 - 2608*n^2 + 5232*n - 2987 + (192*n^3 - 752*n^2 - 668*n + 4331)*x - (192*n^2 + 608*n - 3444)*x^2 - (768*n^3 - 5312*n^2 + 11536*n - 6500)*x^3)*v[i - 3] + (432*n^2 - 1368*n + 927 + (192*n^3 - 1712*n^2 + 3908*n - 3233)*x - (384*n^2 - 1024*n - 1112)*x^2 - (768*n^3 - 6080*n^2 + 14736*n - 10644)*x^3 + (768*n^2 - 3200*n + 2736)*x^4)*deriv(v[i - 3], x) + ((192*n^2 - 608*n + 412)*x - (192*n^2 - 608*n + 796)*x^2 - (768*n^2 - 3200*n + 2352)*x^3 + (768*n^2 - 3200*n + 2736)*x^4)*deriv(deriv(v[i - 3], x), x) - (48*n^2 + 328*n - 417 + (768*n^2 - 4352*n + 4080)*x + (320*n^2 - 1760*n + 2484)*x^2 + (1024*n - 1792)*x^3)*v[i - 2] - (48*n^2 - 152*n + 103 + (384*n - 1664)*x - (192*n^2 - 416*n - 1652)*x^2 + 256*x^3 - 1024*x^4)*deriv(v[i - 2], x) + (512*x^2 - 1024*x^3 + 512*x^4)*deriv(deriv(v[i - 2], x), x) + (96*n + 56 + (256*n - 896)*x + 256*x^2)*v[i - 1])/(32)); 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-723ec2078be12613","steps":["D+1"],"edges":["e-b2371493f3bfa70e"],"text":"A144441 \u2192 D+1"},"headline":"A144441 \u2192 D+1","chain":null}