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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":"3de407d029e6412ef1e2f17b73944ccf09232c7e63356bc365f43bcafbda52f5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-d5c14b9fec875ecc","role":"sequence","name":"Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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:   1;\n  1:   2,     4;\n  2:   4,    20,     9;\n  3:   8,    64,    72,     16;\n  4:  16,   176,   333,    176,     25;\n  5:  32,   448,  1206,   1072,    350,     36;\n  6:  64,  1088,  3816,   4880,   2650,    612,    49;\n  7: 128,  2560, 11088,  18688,  14600,   5544,   980,    64;\n  8: 256,  5888, 30384,  63872,  66025,  35928, 10339,  1472,   81;\n  9: 512, 13312, 79776, 201472, 261050, 187956, 77126, 17728, 2106, 100;\n","recurrence_sentence":"The row polynomials satisfy -(32*n^2+32*n+(72*n^3-32*n^2-96*n)*x+(18*n^4-42*n^3-120*n^2)*x^2-(9*n^4+39*n^3+16*n^2+16*n)*x^3+(3*n^3+4*n^2)*x^4)*P(n,x) = -(64*n^2+64*n+(144*n^3+320*n^2-64*n-256)*x+(36*n^4+276*n^3-616*n-208)*x^2+(36*n^4+48*n^3-284*n^2-392*n-144)*x^3-(39*n^3+121*n^2+106*n+40)*x^4+(6*n^2+14*n+8)*x^5)*P(n-1,x) + ((64*n+64)*x+(144*n^2+384*n+192)*x^2+(36*n^3+420*n^2+632*n+176)*x^3+(72*n^3+288*n^2+244*n+64)*x^4+(9*n^3+9*n^2-30*n-24)*x^5-(3*n^2+7*n+4)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(64*n^2 + 64*n + (144*n^3 + 320*n^2 - 64*n - 256)*x + (36*n^4 + 276*n^3 - 616*n - 208)*x^2 + (36*n^4 + 48*n^3 - 284*n^2 - 392*n - 144)*x^3 - (39*n^3 + 121*n^2 + 106*n + 40)*x^4 + (6*n^2 + 14*n + 8)*x^5)*P[n - 1, x] + ((64*n + 64)*x + (144*n^2 + 384*n + 192)*x^2 + (36*n^3 + 420*n^2 + 632*n + 176)*x^3 + (72*n^3 + 288*n^2 + 244*n + 64)*x^4 + (9*n^3 + 9*n^2 - 30*n - 24)*x^5 - (3*n^2 + 7*n + 4)*x^6)*D[P[n - 1, x], x])/(-32*n^2 - 32*n - (72*n^3 - 32*n^2 - 96*n)*x - (18*n^4 - 42*n^3 - 120*n^2)*x^2 + (9*n^4 + 39*n^3 + 16*n^2 + 16*n)*x^3 - (3*n^3 + 4*n^2)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(64*n^2 + 64*n + (144*n^3 + 320*n^2 - 64*n - 256)*x + (36*n^4 + 276*n^3 - 616*n - 208)*x^2 + (36*n^4 + 48*n^3 - 284*n^2 - 392*n - 144)*x^3 - (39*n^3 + 121*n^2 + 106*n + 40)*x^4 + (6*n^2 + 14*n + 8)*x^5)*v[i - 1] + ((64*n + 64)*x + (144*n^2 + 384*n + 192)*x^2 + (36*n^3 + 420*n^2 + 632*n + 176)*x^3 + (72*n^3 + 288*n^2 + 244*n + 64)*x^4 + (9*n^3 + 9*n^2 - 30*n - 24)*x^5 - (3*n^2 + 7*n + 4)*x^6)*deriv(v[i - 1], x))/(-32*n^2 - 32*n - (72*n^3 - 32*n^2 - 96*n)*x - (18*n^4 - 42*n^3 - 120*n^2)*x^2 + (9*n^4 + 39*n^3 + 16*n^2 + 16*n)*x^3 - (3*n^3 + 4*n^2)*x^4)); 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-d5c14b9fec875ecc","steps":["(\u03b8+1)\u00b2"],"edges":["e-01576e9f8151fd70"],"text":"A121574 \u2192 (\u03b8+1)\u00b2"},"headline":"A121574 \u2192 (\u03b8+1)\u00b2","chain":null}