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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":"fa5000d35bd2836623f26315f879b7bc96b3776e3fa4c9dd4c145e23bbe87929","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-92be85f47bd42a26","role":"sequence","name":"Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.","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:   2;\n   2:   8,    32;\n   3:  20,   224,     180;\n   4:  40,   864,    1944,      640;\n   5:  70,  2464,   10692,     9856,     1750;\n   6: 112,  5824,   41184,    73216,    36400,     4032;\n   7: 168, 12096,  126360,   366080,   351000,   108864,     8232;\n   8: 240, 22848,  330480,  1414400,  2210000,  1321920,   279888,    15360;\n   9: 330, 40128,  767448,  4547840, 10497500, 10232640,  4178328,   642048,   26730;\n  10: 440, 66528, 1625184, 12733952, 40698000, 58605120, 38997728, 11556864, 1347192, 44000;\n","recurrence_sentence":"The row polynomials satisfy (n-1+(2*n^3-7*n^2+5*n)*x+(4*n^5-8*n^4+n^3+5*n^2-n-1)*x^2)*P(n,x) = (n+2+(2*n^3-n^2+2*n-8)*x+(4*n^5+4*n^4-5*n^3-27*n^2+7*n+10)*x^2+(12*n^5+12*n^4-25*n^3-4*n^2+26*n-12)*x^3)*P(n-1,x) + (2*x+(4*n^2-4*n-6)*x^2+(8*n^4+16*n^3-6*n^2-6*n+10)*x^3-(8*n^4+16*n^3-2*n^2-10*n+6)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.","mathematica":"P[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 2 + (2*n^3 - n^2 + 2*n - 8)*x + (4*n^5 + 4*n^4 - 5*n^3 - 27*n^2 + 7*n + 10)*x^2 + (12*n^5 + 12*n^4 - 25*n^3 - 4*n^2 + 26*n - 12)*x^3)*P[n - 1, x] + (2*x + (4*n^2 - 4*n - 6)*x^2 + (8*n^4 + 16*n^3 - 6*n^2 - 6*n + 10)*x^3 - (8*n^4 + 16*n^3 - 2*n^2 - 10*n + 6)*x^4)*D[P[n - 1, x], x])/(n - 1 + (2*n^3 - 7*n^2 + 5*n)*x + (4*n^5 - 8*n^4 + n^3 + 5*n^2 - n - 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((n + 2 + (2*n^3 - n^2 + 2*n - 8)*x + (4*n^5 + 4*n^4 - 5*n^3 - 27*n^2 + 7*n + 10)*x^2 + (12*n^5 + 12*n^4 - 25*n^3 - 4*n^2 + 26*n - 12)*x^3)*v[i - 1] + (2*x + (4*n^2 - 4*n - 6)*x^2 + (8*n^4 + 16*n^3 - 6*n^2 - 6*n + 10)*x^3 - (8*n^4 + 16*n^3 - 2*n^2 - 10*n + 6)*x^4)*deriv(v[i - 1], x))/(n - 1 + (2*n^3 - 7*n^2 + 5*n)*x + (4*n^5 - 8*n^4 + n^3 + 5*n^2 - n - 1)*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-92be85f47bd42a26","steps":["(\u03b8+1)\u00b2"],"edges":["e-3d163da4a898a5ae"],"text":"A375853 \u2192 (\u03b8+1)\u00b2"},"headline":"A375853 \u2192 (\u03b8+1)\u00b2","chain":null}