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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":"bf8bc196cfbbbc3c8aeee405cc7a0c27fc0476271e0b07b67c292fbf0a9eb18e","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:       5,        20;\n  2:      25,       240,        225;\n  3:     125,      2140,       4815,        2000;\n  4:     625,     16840,      67410,       67360,        15625;\n  5:    3125,    123540,     774990,     1377760,       772125,       112500;\n  6:   15625,    866240,    7926615,    22044160,     22018375,      7796160,       765625;\n  7:   78125,   5884940,   74976705,   303179120,    473717375,    299906820,     72090515,     5000000;\n  8:  390625,  39059640,  670489200,  3758352160,   8526912750,   8456292360,   3650441200,   624954240,   31640625;\n  9: 1953125, 254670340, 5748034680, 43178606080, 135466467250, 195071712840, 132234481120, 40874913280, 5157074385, 195312500;\n","recurrence_sentence":"The row polynomials satisfy (n^2+8*n+16)*P(n,x) = -(20*n^2+180*n+400+(20*n^3+240*n^2+956*n+1248)*x-(9*n^4+142*n^3+833*n^2+2152*n+2064)*x^2)*P(n-2,x) - ((10*n^2+90*n+224)*x-(20*n^3+264*n^2+1157*n+1704)*x^2-(4*n^4+50*n^3+203*n^2+248*n-80)*x^3)*d/dx P(n-2,x) - ((n^2+9*n+24)*x^2-(4*n^3+52*n^2+225*n+328)*x^3+(4*n^3+51*n^2+216*n+304)*x^4)*d^2/dx^2 P(n-2,x) + (9*n^2+76*n+160+(4*n^3+48*n^2+192*n+256)*x)*P(n-1,x) + ((2*n^2+17*n+40)*x-(4*n^3+52*n^2+224*n+320)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 20*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 20*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(20*n^2 + 180*n + 400 + (20*n^3 + 240*n^2 + 956*n + 1248)*x - (9*n^4 + 142*n^3 + 833*n^2 + 2152*n + 2064)*x^2)*P[n - 2, x] - ((10*n^2 + 90*n + 224)*x - (20*n^3 + 264*n^2 + 1157*n + 1704)*x^2 - (4*n^4 + 50*n^3 + 203*n^2 + 248*n - 80)*x^3)*D[P[n - 2, x], x] - ((n^2 + 9*n + 24)*x^2 - (4*n^3 + 52*n^2 + 225*n + 328)*x^3 + (4*n^3 + 51*n^2 + 216*n + 304)*x^4)*D[P[n - 2, x], {x, 2}] + (9*n^2 + 76*n + 160 + (4*n^3 + 48*n^2 + 192*n + 256)*x)*P[n - 1, x] + ((2*n^2 + 17*n + 40)*x - (4*n^3 + 52*n^2 + 224*n + 320)*x^2)*D[P[n - 1, x], x])/(n^2 + 8*n + 16)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 20*x; for(n = 2, N, my(i = n + 1); v[i] = (-(20*n^2 + 180*n + 400 + (20*n^3 + 240*n^2 + 956*n + 1248)*x - (9*n^4 + 142*n^3 + 833*n^2 + 2152*n + 2064)*x^2)*v[i - 2] - ((10*n^2 + 90*n + 224)*x - (20*n^3 + 264*n^2 + 1157*n + 1704)*x^2 - (4*n^4 + 50*n^3 + 203*n^2 + 248*n - 80)*x^3)*deriv(v[i - 2], x) - ((n^2 + 9*n + 24)*x^2 - (4*n^3 + 52*n^2 + 225*n + 328)*x^3 + (4*n^3 + 51*n^2 + 216*n + 304)*x^4)*deriv(deriv(v[i - 2], x), x) + (9*n^2 + 76*n + 160 + (4*n^3 + 48*n^2 + 192*n + 256)*x)*v[i - 1] + ((2*n^2 + 17*n + 40)*x - (4*n^3 + 52*n^2 + 224*n + 320)*x^2)*deriv(v[i - 1], x))/(n^2 + 8*n + 16)); 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":["(\u03b8+1)\u00b2"],"edges":["e-658d812c1e72b98e"],"text":"A257607 \u2192 (\u03b8+1)\u00b2"},"headline":"A257607 \u2192 (\u03b8+1)\u00b2","chain":null}