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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":"5f27d0aff56b9f7db4505ae4a4baa536ddf1879e5401a6fd12d16eb578420855","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6a690be6f27c7004","role":"sequence","name":"veronese parent of Binomial coefficients C(n,k) with n-k even, 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: 0,  4;\n  1: 1,  4;\n  2: 0,  4,  18,   16;\n  3: 1, 12,  27,   16;\n  4: 0,  4,  36,   96,  100,   36;\n  5: 1, 20,  90,  160,  125,   36;\n  6: 0,  4,  54,  240,  500,  540,  294,   64;\n  7: 1, 28, 189,  560,  875,  756,  343,   64;\n  8: 0,  4,  72,  448, 1400, 2520, 2744, 1792, 648, 100;\n  9: 1, 36, 324, 1344, 3150, 4536, 4116, 2304, 729, 100;\n","recurrence_sentence":"The row polynomials satisfy (170*n^2-26*n-144+(108*n^3-64*n^2-80*n+36)*x)*P(n,x) = -(197*n^3-259*n^2-82*n+144+(128*n^4-58*n^3-628*n^2+18*n+540)*x+(44*n^5+55*n^4-493*n^3-578*n^2+1116*n-144)*x^2+(8*n^6+8*n^5-60*n^4-56*n^3+128*n^2-28*n)*x^3-(7*n^5-42*n^4-93*n^3+56*n^2+72*n)*x^4)*P(n-1,x) + (56*n^2-38*n-18+(52*n^3+200*n^2-48*n-204)*x+(56*n^4+146*n^3+162*n^2-508*n+144)*x^2+(16*n^5+72*n^4+110*n^3+92*n^2-434*n+144)*x^3+(16*n^5+2*n^4+100*n^3+260*n^2-48*n-330)*x^4-(14*n^4-84*n^3-186*n^2+112*n+144)*x^5)*d/dx P(n-1,x) - (3*n-4-(28*n^2-24*n-40)*x+(12*n^3-5*n^2+235*n+140)*x^2+(8*n^4+40*n^3+166*n^2+632*n+302)*x^3+(16*n^4+37*n^3+270*n^2+768*n+436)*x^4+(8*n^4+2*n^3+162*n^2+478*n+302)*x^5-(7*n^3-35*n^2-128*n-72)*x^6)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 4*x, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 4*x;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(197*n^3 - 259*n^2 - 82*n + 144 + (128*n^4 - 58*n^3 - 628*n^2 + 18*n + 540)*x + (44*n^5 + 55*n^4 - 493*n^3 - 578*n^2 + 1116*n - 144)*x^2 + (8*n^6 + 8*n^5 - 60*n^4 - 56*n^3 + 128*n^2 - 28*n)*x^3 - (7*n^5 - 42*n^4 - 93*n^3 + 56*n^2 + 72*n)*x^4)*P[n - 1, x] + (56*n^2 - 38*n - 18 + (52*n^3 + 200*n^2 - 48*n - 204)*x + (56*n^4 + 146*n^3 + 162*n^2 - 508*n + 144)*x^2 + (16*n^5 + 72*n^4 + 110*n^3 + 92*n^2 - 434*n + 144)*x^3 + (16*n^5 + 2*n^4 + 100*n^3 + 260*n^2 - 48*n - 330)*x^4 - (14*n^4 - 84*n^3 - 186*n^2 + 112*n + 144)*x^5)*D[P[n - 1, x], x] - (3*n - 4 - (28*n^2 - 24*n - 40)*x + (12*n^3 - 5*n^2 + 235*n + 140)*x^2 + (8*n^4 + 40*n^3 + 166*n^2 + 632*n + 302)*x^3 + (16*n^4 + 37*n^3 + 270*n^2 + 768*n + 436)*x^4 + (8*n^4 + 2*n^3 + 162*n^2 + 478*n + 302)*x^5 - (7*n^3 - 35*n^2 - 128*n - 72)*x^6)*D[P[n - 1, x], {x, 2}])/(170*n^2 - 26*n - 144 + (108*n^3 - 64*n^2 - 80*n + 36)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4*x; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(197*n^3 - 259*n^2 - 82*n + 144 + (128*n^4 - 58*n^3 - 628*n^2 + 18*n + 540)*x + (44*n^5 + 55*n^4 - 493*n^3 - 578*n^2 + 1116*n - 144)*x^2 + (8*n^6 + 8*n^5 - 60*n^4 - 56*n^3 + 128*n^2 - 28*n)*x^3 - (7*n^5 - 42*n^4 - 93*n^3 + 56*n^2 + 72*n)*x^4)*v[i - 1] + (56*n^2 - 38*n - 18 + (52*n^3 + 200*n^2 - 48*n - 204)*x + (56*n^4 + 146*n^3 + 162*n^2 - 508*n + 144)*x^2 + (16*n^5 + 72*n^4 + 110*n^3 + 92*n^2 - 434*n + 144)*x^3 + (16*n^5 + 2*n^4 + 100*n^3 + 260*n^2 - 48*n - 330)*x^4 - (14*n^4 - 84*n^3 - 186*n^2 + 112*n + 144)*x^5)*deriv(v[i - 1], x) - (3*n - 4 - (28*n^2 - 24*n - 40)*x + (12*n^3 - 5*n^2 + 235*n + 140)*x^2 + (8*n^4 + 40*n^3 + 166*n^2 + 632*n + 302)*x^3 + (16*n^4 + 37*n^3 + 270*n^2 + 768*n + 436)*x^4 + (8*n^4 + 2*n^3 + 162*n^2 + 478*n + 302)*x^5 - (7*n^3 - 35*n^2 - 128*n - 72)*x^6)*deriv(deriv(v[i - 1], x), x))/(170*n^2 - 26*n - 144 + (108*n^3 - 64*n^2 - 80*n + 36)*x)); 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-6a690be6f27c7004","steps":["(\u03b8+1)\u00b2"],"edges":["e-3d2a41aa1bb0f5ab"],"text":"Veronese parent of A109446 \u2192 (\u03b8+1)\u00b2"},"headline":"Veronese parent of A109446 \u2192 (\u03b8+1)\u00b2","chain":null}