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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":"064bc0ecb98a81009b14bbfd5cbc989cc9f2ea3926b1214cd9783e33e5d89ff7","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-817c504263a54deb","role":"sequence","name":"reversal of Coefficient triangle of certain polynomials N(5; m,x).","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:    6,      4;\n  2:   21,     56,       9;\n  3:   56,    336,     216,      16;\n  4:  126,   1344,    1944,     576,      25;\n  5:  252,   4200,   10800,    7200,    1250,      36;\n  6:  462,  11088,   44550,   52800,   20625,    2376,      49;\n  7:  792,  25872,  149688,  277200,  192500,   49896,    4116,     64;\n  8: 1287,  54912,  432432, 1153152, 1251250,  576576,  107016,   6656,    81;\n  9: 2002, 108108, 1111968, 4036032, 6306300, 4540536, 1498224, 209664, 10206, 100;\n","recurrence_sentence":"The row polynomials satisfy (16*n^2+80*n+(11*n^4+47*n^3-40*n^2)*x+(n^6+5*n^5)*x^2)*P(n,x) = (16*n^2+160*n+400+(11*n^4+102*n^3+387*n^2+312*n-320)*x+(n^6+10*n^5+79*n^4+168*n^3-26*n^2-164*n+40)*x^2+(3*n^6+16*n^5+24*n^4-4*n^3-15*n^2)*x^3)*P(n-1,x) + ((32*n+80)*x+(22*n^3+72*n^2+30*n-100)*x^2+(2*n^5+11*n^4+2*n^3-57*n^2-62*n+20)*x^3-(2*n^5+11*n^4+24*n^3+15*n^2)*x^4)*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[((16*n^2 + 160*n + 400 + (11*n^4 + 102*n^3 + 387*n^2 + 312*n - 320)*x + (n^6 + 10*n^5 + 79*n^4 + 168*n^3 - 26*n^2 - 164*n + 40)*x^2 + (3*n^6 + 16*n^5 + 24*n^4 - 4*n^3 - 15*n^2)*x^3)*P[n - 1, x] + ((32*n + 80)*x + (22*n^3 + 72*n^2 + 30*n - 100)*x^2 + (2*n^5 + 11*n^4 + 2*n^3 - 57*n^2 - 62*n + 20)*x^3 - (2*n^5 + 11*n^4 + 24*n^3 + 15*n^2)*x^4)*D[P[n - 1, x], x])/(16*n^2 + 80*n + (11*n^4 + 47*n^3 - 40*n^2)*x + (n^6 + 5*n^5)*x^2)]];\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] = ((16*n^2 + 160*n + 400 + (11*n^4 + 102*n^3 + 387*n^2 + 312*n - 320)*x + (n^6 + 10*n^5 + 79*n^4 + 168*n^3 - 26*n^2 - 164*n + 40)*x^2 + (3*n^6 + 16*n^5 + 24*n^4 - 4*n^3 - 15*n^2)*x^3)*v[i - 1] + ((32*n + 80)*x + (22*n^3 + 72*n^2 + 30*n - 100)*x^2 + (2*n^5 + 11*n^4 + 2*n^3 - 57*n^2 - 62*n + 20)*x^3 - (2*n^5 + 11*n^4 + 24*n^3 + 15*n^2)*x^4)*deriv(v[i - 1], x))/(16*n^2 + 80*n + (11*n^4 + 47*n^3 - 40*n^2)*x + (n^6 + 5*n^5)*x^2)); 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-5939aa09e8ea2fc4","steps":["reverse","(\u03b8+1)\u00b2"],"edges":["e-3594e486c04f1742","e-f183e9609e283e08"],"text":"A062190 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A062190 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}