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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":"25805110b40f0cb8ad70f7bad4b60ae205ef5b51084e253d5889d707b918d8af","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-2872b2f2cd7c5725","role":"sequence","name":"Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + 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:    6,      7,       1;\n  3:   15,     42,      17,       1;\n  4:   34,    211,     188,      37,        1;\n  5:   73,    898,    1652,     701,       77,       1;\n  6:  152,   3415,   11811,    9921,     2387,     157,       1;\n  7:  311,  12058,   72309,  110725,    51515,    7734,     317,     1;\n  8:  630,  40515,  396311, 1024828,   852123,  245114,   24346,   637,    1;\n  9: 1269, 131610, 2007068, 8243683, 11394576, 5782890, 1103496, 75315, 1277, 1;\n","recurrence_sentence":"The row polynomials satisfy 8*P(n,x) = (6*n^2-10*n-4+(3*n^2-5*n-14)*x-14*x^2-(3*n^2-8*n)*x^3)*P(n-4,x) + ((3*n^2-5*n-2)*x-6*x^2-(3*n^2-8*n)*x^3)*d/dx P(n-4,x) + (6*n^3-25*n^2+21*n+10+(3*n^3-5*n^2-26*n+38)*x-(3*n^2+14*n-48)*x^2-(3*n^3-14*n^2+16*n+16)*x^3)*P(n-3,x) + (9*n^2-15*n-6+(3*n^3-20*n^2+23*n-8)*x-(6*n^2-7*n-32)*x^2-(3*n^3-17*n^2+24*n-6)*x^3+(3*n^2-8*n)*x^4)*d/dx P(n-3,x) + ((3*n^2-5*n-2)*x-(3*n^2-5*n+4)*x^2-(3*n^2-8*n-6)*x^3+(3*n^2-8*n)*x^4)*d^2/dx^2 P(n-3,x) - (3*n^2+7*n+6+(12*n^2-38*n-42)*x+(5*n^2-20*n+36)*x^2+(16*n-16)*x^3)*P(n-2,x) - (3*n^2-5*n-2+(6*n-26)*x-(3*n^2-2*n-32)*x^2+4*x^3-16*x^4)*d/dx P(n-2,x) + (8*x^2-16*x^3+8*x^4)*d^2/dx^2 P(n-2,x) + (6*n+20+(16*n-44)*x+16*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 6 + 7*x + x^2, P(3,x) = 15 + 42*x + 17*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 6 + 7*x + x^2;\nP[3, x_] := 15 + 42*x + 17*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n^2 - 10*n - 4 + (3*n^2 - 5*n - 14)*x - 14*x^2 - (3*n^2 - 8*n)*x^3)*P[n - 4, x] + ((3*n^2 - 5*n - 2)*x - 6*x^2 - (3*n^2 - 8*n)*x^3)*D[P[n - 4, x], x] + (6*n^3 - 25*n^2 + 21*n + 10 + (3*n^3 - 5*n^2 - 26*n + 38)*x - (3*n^2 + 14*n - 48)*x^2 - (3*n^3 - 14*n^2 + 16*n + 16)*x^3)*P[n - 3, x] + (9*n^2 - 15*n - 6 + (3*n^3 - 20*n^2 + 23*n - 8)*x - (6*n^2 - 7*n - 32)*x^2 - (3*n^3 - 17*n^2 + 24*n - 6)*x^3 + (3*n^2 - 8*n)*x^4)*D[P[n - 3, x], x] + ((3*n^2 - 5*n - 2)*x - (3*n^2 - 5*n + 4)*x^2 - (3*n^2 - 8*n - 6)*x^3 + (3*n^2 - 8*n)*x^4)*D[P[n - 3, x], {x, 2}] - (3*n^2 + 7*n + 6 + (12*n^2 - 38*n - 42)*x + (5*n^2 - 20*n + 36)*x^2 + (16*n - 16)*x^3)*P[n - 2, x] - (3*n^2 - 5*n - 2 + (6*n - 26)*x - (3*n^2 - 2*n - 32)*x^2 + 4*x^3 - 16*x^4)*D[P[n - 2, x], x] + (8*x^2 - 16*x^3 + 8*x^4)*D[P[n - 2, x], {x, 2}] + (6*n + 20 + (16*n - 44)*x + 16*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] = 6 + 7*x + x^2; v[4] = 15 + 42*x + 17*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((6*n^2 - 10*n - 4 + (3*n^2 - 5*n - 14)*x - 14*x^2 - (3*n^2 - 8*n)*x^3)*v[i - 4] + ((3*n^2 - 5*n - 2)*x - 6*x^2 - (3*n^2 - 8*n)*x^3)*deriv(v[i - 4], x) + (6*n^3 - 25*n^2 + 21*n + 10 + (3*n^3 - 5*n^2 - 26*n + 38)*x - (3*n^2 + 14*n - 48)*x^2 - (3*n^3 - 14*n^2 + 16*n + 16)*x^3)*v[i - 3] + (9*n^2 - 15*n - 6 + (3*n^3 - 20*n^2 + 23*n - 8)*x - (6*n^2 - 7*n - 32)*x^2 - (3*n^3 - 17*n^2 + 24*n - 6)*x^3 + (3*n^2 - 8*n)*x^4)*deriv(v[i - 3], x) + ((3*n^2 - 5*n - 2)*x - (3*n^2 - 5*n + 4)*x^2 - (3*n^2 - 8*n - 6)*x^3 + (3*n^2 - 8*n)*x^4)*deriv(deriv(v[i - 3], x), x) - (3*n^2 + 7*n + 6 + (12*n^2 - 38*n - 42)*x + (5*n^2 - 20*n + 36)*x^2 + (16*n - 16)*x^3)*v[i - 2] - (3*n^2 - 5*n - 2 + (6*n - 26)*x - (3*n^2 - 2*n - 32)*x^2 + 4*x^3 - 16*x^4)*deriv(v[i - 2], x) + (8*x^2 - 16*x^3 + 8*x^4)*deriv(deriv(v[i - 2], x), x) + (6*n + 20 + (16*n - 44)*x + 16*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-2872b2f2cd7c5725","steps":["D+1"],"edges":["e-aa88e1a16f7cc7e2"],"text":"A144438 \u2192 D+1"},"headline":"A144438 \u2192 D+1","chain":null}