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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":"73d2a30da6baa1ab5ccd1b5c8bcaf93c2d14045d4462bdb9f63f63ab4170c5ce","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-8919a5e7419ba068","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 5.","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:       10,          5;\n  2:       85,        110,         25;\n  3:      660,       1605,        910,         125;\n  4:     4835,      19190,      20120,        6710,         625;\n  5:    34010,     203105,     344440,      209650,       46510,        3125;\n  6:   232185,    1978030,    5014015,     4900700,     1963535,      310310,      15625;\n  7:  1549360,   18132725,   65176830,    94743475,    60602420,    17158155,    2018110,      78125;\n  8: 10155535,  158762510,  779189830,  1599203050,  1515561560,   681889810,  142853170,   12889910,   390625;\n  9: 65620710, 1341008625, 8734659160, 24373297640, 32511952140, 21610635970, 7169356520, 1148011200, 81245710, 1953125;\n","recurrence_sentence":"The row polynomials satisfy (n+2+2*x)*P(n,x) = -(30*n+90+(10*n^2+45*n+105)*x+(n^3+7*n^2+40*n+54)*x^2+(3*n^2+14*n+25)*x^3+(2*n+6)*x^4)*P(n-2,x) - ((5*n+15)*x+(n^2+1)*x^2-(n^2+2*n+9)*x^3-(3*n+5)*x^4-2*x^5)*d/dx P(n-2,x) + (11*n+27+(2*n^2+9*n+32)*x+(5*n+11)*x^2+2*x^3)*P(n-1,x) + ((n+2)*x-n*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 10 + 5*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 10 + 5*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(30*n + 90 + (10*n^2 + 45*n + 105)*x + (n^3 + 7*n^2 + 40*n + 54)*x^2 + (3*n^2 + 14*n + 25)*x^3 + (2*n + 6)*x^4)*P[n - 2, x] - ((5*n + 15)*x + (n^2 + 1)*x^2 - (n^2 + 2*n + 9)*x^3 - (3*n + 5)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (11*n + 27 + (2*n^2 + 9*n + 32)*x + (5*n + 11)*x^2 + 2*x^3)*P[n - 1, x] + ((n + 2)*x - n*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n + 2 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 10 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(30*n + 90 + (10*n^2 + 45*n + 105)*x + (n^3 + 7*n^2 + 40*n + 54)*x^2 + (3*n^2 + 14*n + 25)*x^3 + (2*n + 6)*x^4)*v[i - 2] - ((5*n + 15)*x + (n^2 + 1)*x^2 - (n^2 + 2*n + 9)*x^3 - (3*n + 5)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (11*n + 27 + (2*n^2 + 9*n + 32)*x + (5*n + 11)*x^2 + 2*x^3)*v[i - 1] + ((n + 2)*x - n*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n + 2 + 2*x)); 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-8919a5e7419ba068","steps":["D+1"],"edges":["e-856641ac9f1d35dd"],"text":"A257607 \u2192 D+1"},"headline":"A257607 \u2192 D+1","chain":null}