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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":"b42c45fa5bc223e4896a59b66a4451e13da5c332711ed37ffdec1ea294032aa8","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-36098d4730252f73","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(2*n,2*k)*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:    1,      4;\n   3:    2,     24,        9;\n   4:    5,    120,      135,       16;\n   5:   14,    560,     1260,      448,       25;\n   6:   42,   2520,     9450,     6720,     1125,       36;\n   7:  132,  11088,    62370,    73920,    24750,     2376,      49;\n   8:  429,  48048,   378378,   672672,   375375,    72072,    4459,     64;\n   9: 1430, 205920,  2162160,  5381376,  4504500,  1441440,  178360,   7680,    81;\n  10: 4862, 875160, 11814660, 39207168, 45945900, 22054032, 4548180, 391680, 12393, 100;\n","recurrence_sentence":"The row polynomials satisfy (72*n^2-36*n-(44*n^4-90*n^3+46*n^2-6*n)*x+(2*n^6-5*n^5+4*n^4-n^3)*x^2)*P(n,x) = (288*n^2-576*n+216-(176*n^4-624*n^3-140*n^2+1428*n-828)*x+(8*n^6-32*n^5-182*n^4+496*n^3-168*n^2-350*n+228)*x^2+(6*n^6-8*n^5-12*n^4+24*n^3-3*n^2-13*n+6)*x^3)*P(n-1,x) + ((576*n-432)*x-(352*n^3-280*n^2-152*n+156)*x^2+(16*n^5+4*n^4+48*n^3-54*n^2-54*n+54)*x^3-(4*n^5+n^4-10*n^3+4*n^2+5*n-3)*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[((288*n^2 - 576*n + 216 - (176*n^4 - 624*n^3 - 140*n^2 + 1428*n - 828)*x + (8*n^6 - 32*n^5 - 182*n^4 + 496*n^3 - 168*n^2 - 350*n + 228)*x^2 + (6*n^6 - 8*n^5 - 12*n^4 + 24*n^3 - 3*n^2 - 13*n + 6)*x^3)*P[n - 1, x] + ((576*n - 432)*x - (352*n^3 - 280*n^2 - 152*n + 156)*x^2 + (16*n^5 + 4*n^4 + 48*n^3 - 54*n^2 - 54*n + 54)*x^3 - (4*n^5 + n^4 - 10*n^3 + 4*n^2 + 5*n - 3)*x^4)*D[P[n - 1, x], x])/(72*n^2 - 36*n - (44*n^4 - 90*n^3 + 46*n^2 - 6*n)*x + (2*n^6 - 5*n^5 + 4*n^4 - 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] = ((288*n^2 - 576*n + 216 - (176*n^4 - 624*n^3 - 140*n^2 + 1428*n - 828)*x + (8*n^6 - 32*n^5 - 182*n^4 + 496*n^3 - 168*n^2 - 350*n + 228)*x^2 + (6*n^6 - 8*n^5 - 12*n^4 + 24*n^3 - 3*n^2 - 13*n + 6)*x^3)*v[i - 1] + ((576*n - 432)*x - (352*n^3 - 280*n^2 - 152*n + 156)*x^2 + (16*n^5 + 4*n^4 + 48*n^3 - 54*n^2 - 54*n + 54)*x^3 - (4*n^5 + n^4 - 10*n^3 + 4*n^2 + 5*n - 3)*x^4)*deriv(v[i - 1], x))/(72*n^2 - 36*n - (44*n^4 - 90*n^3 + 46*n^2 - 6*n)*x + (2*n^6 - 5*n^5 + 4*n^4 - 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-36098d4730252f73","steps":["(\u03b8+1)\u00b2"],"edges":["e-766d6dc15435f53f"],"text":"A280580 \u2192 (\u03b8+1)\u00b2"},"headline":"A280580 \u2192 (\u03b8+1)\u00b2","chain":null}