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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":"806237b5fd236a4272a3e0c8cd4fcfbf06555a0bfcd4fcc24c8ca239dc6692d4","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-29bd79a8a49e73a9","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of polynomials u(n,x) jointly 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98026,  679792,  1996612,  3255136,  3239136,  2032128,   797888,   186368,   22784,    1024;\n  8: 282267, 2205338,  7429704, 14198448, 17019632, 13338528,  6892032,  2300416,  467712,   50688,   2048;\n  9: 806089, 7007976, 26654904, 58502176, 82240240, 77625856, 50067584, 21966336, 6372864, 1146880, 111616, 4096;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-140*n-216+(64*n^3+288*n^2+432*n+216)*x-(80*n+96)*x^2-(64*n+96)*x^3)*P(n,x) = (24*n^3+32*n^2-358*n-780+(112*n^3+672*n^2+1100*n+392)*x+(64*n^3+448*n^2+712*n+272)*x^2-(240*n+416)*x^3-(64*n+128)*x^4)*P(n-1,x) + (20*n^2+20*n-135+(128*n^2+368*n-264)*x+(208*n^2+784*n-224)*x^2+(64*n^2+256*n-656)*x^3-400*x^4-64*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 35 + 38*x + 8*x^2.","mathematica":"P[0, x_] := 35 + 38*x + 8*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((24*n^3 + 32*n^2 - 358*n - 780 + (112*n^3 + 672*n^2 + 1100*n + 392)*x + (64*n^3 + 448*n^2 + 712*n + 272)*x^2 - (240*n + 416)*x^3 - (64*n + 128)*x^4)*P[n - 1, x] + (20*n^2 + 20*n - 135 + (128*n^2 + 368*n - 264)*x + (208*n^2 + 784*n - 224)*x^2 + (64*n^2 + 256*n - 656)*x^3 - 400*x^4 - 64*x^5)*D[P[n - 1, x], x])/(16*n^3 - 140*n - 216 + (64*n^3 + 288*n^2 + 432*n + 216)*x - (80*n + 96)*x^2 - (64*n + 96)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 35 + 38*x + 8*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24*n^3 + 32*n^2 - 358*n - 780 + (112*n^3 + 672*n^2 + 1100*n + 392)*x + (64*n^3 + 448*n^2 + 712*n + 272)*x^2 - (240*n + 416)*x^3 - (64*n + 128)*x^4)*v[i - 1] + (20*n^2 + 20*n - 135 + (128*n^2 + 368*n - 264)*x + (208*n^2 + 784*n - 224)*x^2 + (64*n^2 + 256*n - 656)*x^3 - 400*x^4 - 64*x^5)*deriv(v[i - 1], x))/(16*n^3 - 140*n - 216 + (64*n^3 + 288*n^2 + 432*n + 216)*x - (80*n + 96)*x^2 - (64*n + 96)*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-dfab35176bd08e72","steps":["basis A013609","D+1"],"edges":["e-0adfae28f720d284","e-67ea21b7e7881eeb"],"text":"A207606 \u2192 basis A013609 \u2192 D+1"},"headline":"A207606 \u2192 basis A013609 \u2192 D+1","chain":null}