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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":"5ee5fa4fe0b632f9e6a100f5e1f1482ec4e0cb476b156b17955164241abe9f3c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b0d6ec97a8d172b8","role":"sequence","name":"Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0,  4;\n   3: 0,  8,   27;\n   4: 0, 12,  108,   160;\n   5: 0, 16,  270,   960,    875;\n   6: 0, 20,  540,  3360,   7000,    4536;\n   7: 0, 24,  945,  8960,  31500,   45360,   22638;\n   8: 0, 28, 1512, 20160, 105000,  249480,  271656,   109824;\n   9: 0, 32, 2268, 40320, 288750,  997920, 1765764,  1537536,  521235;\n  10: 0, 36, 3240, 73920, 693000, 3243240, 8240232, 11531520, 8339760, 2431000;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4-44*n^3+167*n^2-253*n+126)*P(n,x) = -(4*n^4-52*n^3+239*n^2-455*n+300)*P(n-4,x) + (16*n^4-200*n^3+896*n^2-1684*n+1104+(64*n^4-800*n^3+3568*n^2-6648*n+4320)*x)*P(n-3,x) - (64*n^4-800*n^3+3568*n^2-6648*n+4320)*x^2*d/dx P(n-3,x) - (24*n^4-288*n^3+1242*n^2-2268*n+1458+(128*n^4-1536*n^3+6624*n^2-12096*n+7776)*x-(576*n^4-6912*n^3+29664*n^2-53568*n+34020)*x^2)*P(n-2,x) + ((128*n^4-1536*n^3+6624*n^2-12096*n+7776)*x^2+(256*n^4-3072*n^3+13184*n^2-23808*n+15120)*x^3)*d/dx P(n-2,x) + (16*n^4-184*n^3+752*n^2-1292*n+792+(64*n^4-736*n^3+2992*n^2-5064*n+3024)*x)*P(n-1,x) - (64*n^4-736*n^3+2992*n^2-5064*n+3024)*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 4*x, P(3,x) = 8*x + 27*x^2, P(4,x) = 12*x + 108*x^2 + 160*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4*x;\nP[3, x_] := 8*x + 27*x^2;\nP[4, x_] := 12*x + 108*x^2 + 160*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*P[n - 4, x] + (16*n^4 - 200*n^3 + 896*n^2 - 1684*n + 1104 + (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x)*P[n - 3, x] - (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x^2*D[P[n - 3, x], x] - (24*n^4 - 288*n^3 + 1242*n^2 - 2268*n + 1458 + (128*n^4 - 1536*n^3 + 6624*n^2 - 12096*n + 7776)*x - (576*n^4 - 6912*n^3 + 29664*n^2 - 53568*n + 34020)*x^2)*P[n - 2, x] + ((128*n^4 - 1536*n^3 + 6624*n^2 - 12096*n + 7776)*x^2 + (256*n^4 - 3072*n^3 + 13184*n^2 - 23808*n + 15120)*x^3)*D[P[n - 2, x], x] + (16*n^4 - 184*n^3 + 752*n^2 - 1292*n + 792 + (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x)*P[n - 1, x] - (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x^2*D[P[n - 1, x], x])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4*x; v[3] = 8*x + 27*x^2; v[4] = 12*x + 108*x^2 + 160*x^3; for(n = 5, N, my(i = n); v[i] = (-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*v[i - 4] + (16*n^4 - 200*n^3 + 896*n^2 - 1684*n + 1104 + (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x)*v[i - 3] - (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x^2*deriv(v[i - 3], x) - (24*n^4 - 288*n^3 + 1242*n^2 - 2268*n + 1458 + (128*n^4 - 1536*n^3 + 6624*n^2 - 12096*n + 7776)*x - (576*n^4 - 6912*n^3 + 29664*n^2 - 53568*n + 34020)*x^2)*v[i - 2] + ((128*n^4 - 1536*n^3 + 6624*n^2 - 12096*n + 7776)*x^2 + (256*n^4 - 3072*n^3 + 13184*n^2 - 23808*n + 15120)*x^3)*deriv(v[i - 2], x) + (16*n^4 - 184*n^3 + 752*n^2 - 1292*n + 792 + (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x)*v[i - 1] - (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x^2*deriv(v[i - 1], x))/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b0d6ec97a8d172b8","steps":["(\u03b8+1)\u00b2"],"edges":["e-0d261ec0a6ea2dc4"],"text":"A253283 \u2192 (\u03b8+1)\u00b2"},"headline":"A253283 \u2192 (\u03b8+1)\u00b2","chain":null}