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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":"e0ee0b7a4eb8f62a12abddc9524883507b19d53b66a6995d930da20d6217167b","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-14be0ce6edbde2a1","role":"sequence","name":"Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2, read by rows.","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: 1,      4;\n  2: 1,     32,        9;\n  3: 1,    124,      279,        16;\n  4: 1,    408,     2934,      1632,         25;\n  5: 1,   1268,    21654,     38496,       7925,         36;\n  6: 1,   3856,   135783,    549376,     377175,      34704,         49;\n  7: 1,  11628,   780057,   6095152,    9523675,    3120228,     142443,        64;\n  8: 1,  34952,  4257756,  58403296,  173142950,  131407416,   23181116,    559232,      81;\n  9: 1, 104932, 22520736, 509998080, 2581140550, 3716842392, 1561869120, 160147456, 2124873, 100;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-12*n+9)*P(n,x) = (144*n^2-288*n+204)*x^2*P(n-4,x) + (64*n^2-128*n+96)*x^3*d/dx P(n-4,x) - ((32*n^2-64*n-72)*x-(288*n^3-1040*n^2+1128*n-420)*x^2)*P(n-3,x) + ((32*n^2-64*n+72)*x^2+(128*n^3-672*n^2+928*n-504)*x^3)*d/dx P(n-3,x) + ((64*n^2-128*n+96)*x^3-(64*n^2-128*n+96)*x^4)*d^2/dx^2 P(n-3,x) + (4*n^2-8*n+3-(32*n^3-144*n^2+120*n+36)*x+(144*n^4-608*n^3+792*n^2-216*n-135)*x^2)*P(n-2,x) - ((16*n^2-32*n-24)*x-(32*n^3-48*n^2-24*n-84)*x^2-(64*n^4-480*n^3+928*n^2-408*n-180)*x^3)*d/dx P(n-2,x) - ((16*n^2-32*n-12)*x^2-(64*n^3-160*n^2+64*n+24)*x^3+(64*n^3-176*n^2+96*n+36)*x^4)*d^2/dx^2 P(n-2,x) - (4*n-6-(32*n^3-112*n^2+120*n-36)*x)*P(n-1,x) + ((16*n^2-40*n+12)*x-(32*n^3-80*n^2+24*n+36)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 32*x + 9*x^2, P(3,x) = 1 + 124*x + 279*x^2 + 16*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[2, x_] := 1 + 32*x + 9*x^2;\nP[3, x_] := 1 + 124*x + 279*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((144*n^2 - 288*n + 204)*x^2*P[n - 4, x] + (64*n^2 - 128*n + 96)*x^3*D[P[n - 4, x], x] - ((32*n^2 - 64*n - 72)*x - (288*n^3 - 1040*n^2 + 1128*n - 420)*x^2)*P[n - 3, x] + ((32*n^2 - 64*n + 72)*x^2 + (128*n^3 - 672*n^2 + 928*n - 504)*x^3)*D[P[n - 3, x], x] + ((64*n^2 - 128*n + 96)*x^3 - (64*n^2 - 128*n + 96)*x^4)*D[P[n - 3, x], {x, 2}] + (4*n^2 - 8*n + 3 - (32*n^3 - 144*n^2 + 120*n + 36)*x + (144*n^4 - 608*n^3 + 792*n^2 - 216*n - 135)*x^2)*P[n - 2, x] - ((16*n^2 - 32*n - 24)*x - (32*n^3 - 48*n^2 - 24*n - 84)*x^2 - (64*n^4 - 480*n^3 + 928*n^2 - 408*n - 180)*x^3)*D[P[n - 2, x], x] - ((16*n^2 - 32*n - 12)*x^2 - (64*n^3 - 160*n^2 + 64*n + 24)*x^3 + (64*n^3 - 176*n^2 + 96*n + 36)*x^4)*D[P[n - 2, x], {x, 2}] - (4*n - 6 - (32*n^3 - 112*n^2 + 120*n - 36)*x)*P[n - 1, x] + ((16*n^2 - 40*n + 12)*x - (32*n^3 - 80*n^2 + 24*n + 36)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 12*n + 9)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; v[3] = 1 + 32*x + 9*x^2; v[4] = 1 + 124*x + 279*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((144*n^2 - 288*n + 204)*x^2*v[i - 4] + (64*n^2 - 128*n + 96)*x^3*deriv(v[i - 4], x) - ((32*n^2 - 64*n - 72)*x - (288*n^3 - 1040*n^2 + 1128*n - 420)*x^2)*v[i - 3] + ((32*n^2 - 64*n + 72)*x^2 + (128*n^3 - 672*n^2 + 928*n - 504)*x^3)*deriv(v[i - 3], x) + ((64*n^2 - 128*n + 96)*x^3 - (64*n^2 - 128*n + 96)*x^4)*deriv(deriv(v[i - 3], x), x) + (4*n^2 - 8*n + 3 - (32*n^3 - 144*n^2 + 120*n + 36)*x + (144*n^4 - 608*n^3 + 792*n^2 - 216*n - 135)*x^2)*v[i - 2] - ((16*n^2 - 32*n - 24)*x - (32*n^3 - 48*n^2 - 24*n - 84)*x^2 - (64*n^4 - 480*n^3 + 928*n^2 - 408*n - 180)*x^3)*deriv(v[i - 2], x) - ((16*n^2 - 32*n - 12)*x^2 - (64*n^3 - 160*n^2 + 64*n + 24)*x^3 + (64*n^3 - 176*n^2 + 96*n + 36)*x^4)*deriv(deriv(v[i - 2], x), x) - (4*n - 6 - (32*n^3 - 112*n^2 + 120*n - 36)*x)*v[i - 1] + ((16*n^2 - 40*n + 12)*x - (32*n^3 - 80*n^2 + 24*n + 36)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 12*n + 9)); 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-14be0ce6edbde2a1","steps":["(\u03b8+1)\u00b2"],"edges":["e-beda5165fcb4775f"],"text":"A144439 \u2192 (\u03b8+1)\u00b2"},"headline":"A144439 \u2192 (\u03b8+1)\u00b2","chain":null}