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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":"30fa34e943a766abc915f9ec6d751545679a574318b3fa5004cf8be09ac0fe84","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a7d7f0f34831fe03","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(2*n+1, 2*k+1)*binomial(2*n-2*k, n-k)/(n+1-k) for 0 <= k <= 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:     3,       4;\n   3:    10,      40,        9;\n   4:    35,     280,      189,        16;\n   5:   126,    1680,     2268,       576,       25;\n   6:   462,    9240,    20790,     10560,     1375,       36;\n   7:  1716,   48048,   162162,    137280,    35750,     2808,      49;\n   8:  6435,  240240,  1135134,   1441440,   625625,    98280,    5145,     64;\n   9: 24310, 1166880,  7351344,  13069056,  8508500,  2227680,  233240,   8704,    81;\n  10: 92378, 5542680, 44895708, 106419456, 96996900, 38093328, 6647340, 496128, 13851, 100;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2+4*n-(20*n^4-18*n^3-10*n^2+2*n)*x+(2*n^6-3*n^5+n^3)*x^2)*P(n,x) = (32*n^2-8-(80*n^4-112*n^3-100*n^2+156*n-36)*x+(8*n^6-16*n^5-102*n^4+160*n^3+24*n^2-110*n+36)*x^2+(6*n^6-4*n^5-12*n^4+12*n^3+3*n^2-7*n+2)*x^3)*P(n-1,x) + ((64*n-16)*x-(160*n^3-24*n^2-88*n+36)*x^2+(16*n^5+12*n^4+16*n^3-6*n^2-14*n+6)*x^3-(4*n^5+3*n^4-6*n^3+3*n-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((32*n^2 - 8 - (80*n^4 - 112*n^3 - 100*n^2 + 156*n - 36)*x + (8*n^6 - 16*n^5 - 102*n^4 + 160*n^3 + 24*n^2 - 110*n + 36)*x^2 + (6*n^6 - 4*n^5 - 12*n^4 + 12*n^3 + 3*n^2 - 7*n + 2)*x^3)*P[n - 1, x] + ((64*n - 16)*x - (160*n^3 - 24*n^2 - 88*n + 36)*x^2 + (16*n^5 + 12*n^4 + 16*n^3 - 6*n^2 - 14*n + 6)*x^3 - (4*n^5 + 3*n^4 - 6*n^3 + 3*n - 1)*x^4)*D[P[n - 1, x], x])/(8*n^2 + 4*n - (20*n^4 - 18*n^3 - 10*n^2 + 2*n)*x + (2*n^6 - 3*n^5 + n^3)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((32*n^2 - 8 - (80*n^4 - 112*n^3 - 100*n^2 + 156*n - 36)*x + (8*n^6 - 16*n^5 - 102*n^4 + 160*n^3 + 24*n^2 - 110*n + 36)*x^2 + (6*n^6 - 4*n^5 - 12*n^4 + 12*n^3 + 3*n^2 - 7*n + 2)*x^3)*v[i - 1] + ((64*n - 16)*x - (160*n^3 - 24*n^2 - 88*n + 36)*x^2 + (16*n^5 + 12*n^4 + 16*n^3 - 6*n^2 - 14*n + 6)*x^3 - (4*n^5 + 3*n^4 - 6*n^3 + 3*n - 1)*x^4)*deriv(v[i - 1], x))/(8*n^2 + 4*n - (20*n^4 - 18*n^3 - 10*n^2 + 2*n)*x + (2*n^6 - 3*n^5 + n^3)*x^2)); 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-a7d7f0f34831fe03","steps":["(\u03b8+1)\u00b2"],"edges":["e-2eeac86491785187"],"text":"A281000 \u2192 (\u03b8+1)\u00b2"},"headline":"A281000 \u2192 (\u03b8+1)\u00b2","chain":null}