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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":"8aad3f02f99a1041ea3ed0eef656d28520481ac70a2c7f07f4de498de3bb8a0d","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-273f45fc84d8c52d","role":"sequence","name":"basis_map_A013609 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","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:  1,    8;\n   3:  2,   24,    36;\n   4:  3,   56,   180,    128;\n   5:  5,  120,   576,    896,    400;\n   6:  8,  240,  1548,   3712,   3600,    1152;\n   7: 13,  464,  3744,  12160,  18400,   12672,    3136;\n   8: 21,  872,  8460,  34688,  71600,   77184,   40768,   8192;\n   9: 34, 1608, 18216,  90240, 236400,  349056,  288512, 122880,  20736;\n  10: 55, 2920, 37836, 219776, 698400, 1313280, 1489600, 991232, 352512, 51200;\n","recurrence_sentence":"The row polynomials satisfy (25+(150*n-170)*x+(240*n^2-440*n+136)*x^2+(120*n^3-200*n^2-144*n+208)*x^3+(16*n^4-208*n^2+224*n+16)*x^4-(32*n^3-32*n^2-32*n+32)*x^5)*P(n,x) = (25+(150*n+30)*x+(240*n^2+340*n-104)*x^2+(120*n^3+640*n^2-104*n-144)*x^3+(16*n^4+304*n^3+208*n^2-496*n+16)*x^4+(32*n^4+96*n^3-288*n^2-256*n+96)*x^5-(64*n^3+128*n^2)*x^6)*P(n-2,x) + (25+(150*n+30)*x+(240*n^2+200*n-104)*x^2+(120*n^3+520*n^2-544*n-144)*x^3+(16*n^4+272*n^3-96*n^2-768*n+16)*x^4+(32*n^4+32*n^3-384*n^2-160*n+96)*x^5-(64*n^3+64*n^2)*x^6)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 8*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((25 + (150*n + 30)*x + (240*n^2 + 340*n - 104)*x^2 + (120*n^3 + 640*n^2 - 104*n - 144)*x^3 + (16*n^4 + 304*n^3 + 208*n^2 - 496*n + 16)*x^4 + (32*n^4 + 96*n^3 - 288*n^2 - 256*n + 96)*x^5 - (64*n^3 + 128*n^2)*x^6)*P[n - 2, x] + (25 + (150*n + 30)*x + (240*n^2 + 200*n - 104)*x^2 + (120*n^3 + 520*n^2 - 544*n - 144)*x^3 + (16*n^4 + 272*n^3 - 96*n^2 - 768*n + 16)*x^4 + (32*n^4 + 32*n^3 - 384*n^2 - 160*n + 96)*x^5 - (64*n^3 + 64*n^2)*x^6)*P[n - 1, x])/(25 + (150*n - 170)*x + (240*n^2 - 440*n + 136)*x^2 + (120*n^3 - 200*n^2 - 144*n + 208)*x^3 + (16*n^4 - 208*n^2 + 224*n + 16)*x^4 - (32*n^3 - 32*n^2 - 32*n + 32)*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 8*x; for(n = 3, N, my(i = n); v[i] = ((25 + (150*n + 30)*x + (240*n^2 + 340*n - 104)*x^2 + (120*n^3 + 640*n^2 - 104*n - 144)*x^3 + (16*n^4 + 304*n^3 + 208*n^2 - 496*n + 16)*x^4 + (32*n^4 + 96*n^3 - 288*n^2 - 256*n + 96)*x^5 - (64*n^3 + 128*n^2)*x^6)*v[i - 2] + (25 + (150*n + 30)*x + (240*n^2 + 200*n - 104)*x^2 + (120*n^3 + 520*n^2 - 544*n - 144)*x^3 + (16*n^4 + 272*n^3 - 96*n^2 - 768*n + 16)*x^4 + (32*n^4 + 32*n^3 - 384*n^2 - 160*n + 96)*x^5 - (64*n^3 + 64*n^2)*x^6)*v[i - 1])/(25 + (150*n - 170)*x + (240*n^2 - 440*n + 136)*x^2 + (120*n^3 - 200*n^2 - 144*n + 208)*x^3 + (16*n^4 - 208*n^2 + 224*n + 16)*x^4 - (32*n^3 - 32*n^2 - 32*n + 32)*x^5)); 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-392f49b496f79fbf","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-6b05245c845c9a6d","e-f988d6fe8e777ad5"],"text":"A026729 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A026729 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}