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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":"2f94f19f51ea7cb263da19e88f15603346964fed023db96c3d60a43c3edce34b","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-e655e253465c7aff","role":"sequence","name":"veronese parent of Binomial coefficients C(n,k) with n-k odd, 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,   9;\n  1: 1,  8,   9;\n  2: 0,  4,  27,   48,   25;\n  3: 1, 16,  54,   64,   25;\n  4: 0,  4,  45,  160,  250,  180,    49;\n  5: 1, 24, 135,  320,  375,  216,    49;\n  6: 0,  4,  63,  336,  875, 1260,  1029,  448,   81;\n  7: 1, 32, 252,  896, 1750, 2016,  1372,  512,   81;\n  8: 0,  4,  81,  576, 2100, 4536,  6174, 5376, 2916,  900, 121;\n  9: 1, 40, 405, 1920, 5250, 9072, 10290, 7680, 3645, 1000, 121;\n","recurrence_sentence":"The row polynomials satisfy (170*n^2+314*n+(108*n^3+260*n^2+116*n)*x)*P(n,x) = -(197*n^3+332*n^2-9*n+(128*n^4+454*n^3-34*n^2-900*n)*x+(44*n^5+275*n^4+167*n^3-1287*n^2-1079*n)*x^2+(8*n^6+56*n^5+100*n^4-56*n^3-200*n^2-92*n)*x^3-(7*n^5-7*n^4-191*n^3-405*n^2-228*n)*x^4)*P(n-1,x) + (56*n^2+74*n+(52*n^3+356*n^2+508*n)*x+(56*n^4+370*n^3+936*n^2+478*n)*x^2+(16*n^5+152*n^4+558*n^3+1014*n^2+448*n)*x^3+(16*n^5+82*n^4+268*n^3+732*n^2+860*n)*x^4-(14*n^4-28*n^3-354*n^2-456*n)*x^5)*d/dx P(n-1,x) - (3*n-1-(28*n^2+32*n-36)*x+(12*n^3+31*n^2+261*n+382)*x^2+(8*n^4+72*n^3+334*n^2+1116*n+1148)*x^3+(16*n^4+101*n^3+477*n^2+1483*n+1527)*x^4+(8*n^4+34*n^3+216*n^2+840*n+952)*x^5-(7*n^3-14*n^2-177*n-228)*x^6)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 4*x + 9*x^2.","mathematica":"P[0, x_] := 4*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(197*n^3 + 332*n^2 - 9*n + (128*n^4 + 454*n^3 - 34*n^2 - 900*n)*x + (44*n^5 + 275*n^4 + 167*n^3 - 1287*n^2 - 1079*n)*x^2 + (8*n^6 + 56*n^5 + 100*n^4 - 56*n^3 - 200*n^2 - 92*n)*x^3 - (7*n^5 - 7*n^4 - 191*n^3 - 405*n^2 - 228*n)*x^4)*P[n - 1, x] + (56*n^2 + 74*n + (52*n^3 + 356*n^2 + 508*n)*x + (56*n^4 + 370*n^3 + 936*n^2 + 478*n)*x^2 + (16*n^5 + 152*n^4 + 558*n^3 + 1014*n^2 + 448*n)*x^3 + (16*n^5 + 82*n^4 + 268*n^3 + 732*n^2 + 860*n)*x^4 - (14*n^4 - 28*n^3 - 354*n^2 - 456*n)*x^5)*D[P[n - 1, x], x] - (3*n - 1 - (28*n^2 + 32*n - 36)*x + (12*n^3 + 31*n^2 + 261*n + 382)*x^2 + (8*n^4 + 72*n^3 + 334*n^2 + 1116*n + 1148)*x^3 + (16*n^4 + 101*n^3 + 477*n^2 + 1483*n + 1527)*x^4 + (8*n^4 + 34*n^3 + 216*n^2 + 840*n + 952)*x^5 - (7*n^3 - 14*n^2 - 177*n - 228)*x^6)*D[P[n - 1, x], {x, 2}])/(170*n^2 + 314*n + (108*n^3 + 260*n^2 + 116*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(197*n^3 + 332*n^2 - 9*n + (128*n^4 + 454*n^3 - 34*n^2 - 900*n)*x + (44*n^5 + 275*n^4 + 167*n^3 - 1287*n^2 - 1079*n)*x^2 + (8*n^6 + 56*n^5 + 100*n^4 - 56*n^3 - 200*n^2 - 92*n)*x^3 - (7*n^5 - 7*n^4 - 191*n^3 - 405*n^2 - 228*n)*x^4)*v[i - 1] + (56*n^2 + 74*n + (52*n^3 + 356*n^2 + 508*n)*x + (56*n^4 + 370*n^3 + 936*n^2 + 478*n)*x^2 + (16*n^5 + 152*n^4 + 558*n^3 + 1014*n^2 + 448*n)*x^3 + (16*n^5 + 82*n^4 + 268*n^3 + 732*n^2 + 860*n)*x^4 - (14*n^4 - 28*n^3 - 354*n^2 - 456*n)*x^5)*deriv(v[i - 1], x) - (3*n - 1 - (28*n^2 + 32*n - 36)*x + (12*n^3 + 31*n^2 + 261*n + 382)*x^2 + (8*n^4 + 72*n^3 + 334*n^2 + 1116*n + 1148)*x^3 + (16*n^4 + 101*n^3 + 477*n^2 + 1483*n + 1527)*x^4 + (8*n^4 + 34*n^3 + 216*n^2 + 840*n + 952)*x^5 - (7*n^3 - 14*n^2 - 177*n - 228)*x^6)*deriv(deriv(v[i - 1], x), x))/(170*n^2 + 314*n + (108*n^3 + 260*n^2 + 116*n)*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-e655e253465c7aff","steps":["(\u03b8+1)\u00b2"],"edges":["e-47503630d4a049ac"],"text":"Veronese parent of A109447 \u2192 (\u03b8+1)\u00b2"},"headline":"Veronese parent of A109447 \u2192 (\u03b8+1)\u00b2","chain":null}