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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"2523f1e07bb2c7b007c2f1ee4b261b78f14c1c5f8863584cde5b2f3a283815f1","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-30cb3cad216ed189","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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:      4,       4;\n  2:     15,      32,       9;\n  3:     56,     184,     108,      16;\n  4:    209,     928,     837,     256,      25;\n  5:    780,    4364,    5328,    2496,     500,      36;\n  6:   2911,   19648,   30294,   19200,    5875,     864,     49;\n  7:  10864,   85872,  160056,  128160,   53000,   11880,   1372,    64;\n  8:  40545,  367296,  803142,  777984,  406375,  122976,  21609,  2048,   81;\n  9: 151316, 1545492, 3878928, 4411584, 2788100, 1065204, 252448, 36352, 2916, 100;\n","recurrence_sentence":"The row polynomials satisfy (1152*n+96*n*x-(240*n^3-32*n)*x^2-40*n^3*x^3+2*n^5*x^4)*P(n,x) = (2304*n+2304+(1008*n^2+2496*n+1488)*x-(480*n^3+528*n^2-256*n-304)*x^2-(24*n^4+284*n^3+464*n^2+212*n+8)*x^3+(4*n^5+4*n^4-36*n^3-72*n^2-44*n-8)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (1728+1728*x-(1008*n^2+1008*n-432)*x^2-(912*n^2+912*n+64)*x^3+(12*n^4+24*n^3-232*n^2-244*n-40)*x^4+(8*n^4+16*n^3-12*n^2-20*n-4)*x^5+(n^4+2*n^3+n^2)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2304*n + 2304 + (1008*n^2 + 2496*n + 1488)*x - (480*n^3 + 528*n^2 - 256*n - 304)*x^2 - (24*n^4 + 284*n^3 + 464*n^2 + 212*n + 8)*x^3 + (4*n^5 + 4*n^4 - 36*n^3 - 72*n^2 - 44*n - 8)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (1728 + 1728*x - (1008*n^2 + 1008*n - 432)*x^2 - (912*n^2 + 912*n + 64)*x^3 + (12*n^4 + 24*n^3 - 232*n^2 - 244*n - 40)*x^4 + (8*n^4 + 16*n^3 - 12*n^2 - 20*n - 4)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(1152*n + 96*n*x - (240*n^3 - 32*n)*x^2 - 40*n^3*x^3 + 2*n^5*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2304*n + 2304 + (1008*n^2 + 2496*n + 1488)*x - (480*n^3 + 528*n^2 - 256*n - 304)*x^2 - (24*n^4 + 284*n^3 + 464*n^2 + 212*n + 8)*x^3 + (4*n^5 + 4*n^4 - 36*n^3 - 72*n^2 - 44*n - 8)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (1728 + 1728*x - (1008*n^2 + 1008*n - 432)*x^2 - (912*n^2 + 912*n + 64)*x^3 + (12*n^4 + 24*n^3 - 232*n^2 - 244*n - 40)*x^4 + (8*n^4 + 16*n^3 - 12*n^2 - 20*n - 4)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(1152*n + 96*n*x - (240*n^3 - 32*n)*x^2 - 40*n^3*x^3 + 2*n^5*x^4)); 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-30cb3cad216ed189","steps":["(\u03b8+1)\u00b2"],"edges":["e-9b17678c30419d79"],"text":"A207823 \u2192 (\u03b8+1)\u00b2"},"headline":"A207823 \u2192 (\u03b8+1)\u00b2","chain":null}