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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":"43a1d70c672ca4f636b98db0b032cf4430beeadf090d682d19b043fe841c52ca","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-08fe62d9c34c34e9","role":"sequence","name":"basis_map_A013610 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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:      4,       24;\n  2:     16,      192,       243;\n  3:     64,     1152,      3240,      1728;\n  4:    256,     6144,     27216,     34560,      10125;\n  5:   1024,    30720,    186624,    387072,     283500,      52488;\n  6:   4096,   147456,   1140480,   3317760,    4082400,    1959552,     250047;\n  7:  16384,   688128,   6469632,  24330240,   42768000,   35271936,   12002256,    1119744;\n  8:  65536,  3145728,  34836480, 161021952,  370656000,  443418624,  264049632,   67184640,   4782969;\n  9: 262144, 14155776, 180486144, 990904320, 2830464000, 4483454976, 3923023104, 1773674496, 350751060, 19683000;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3+16*n^2+4*n+(72*n^4+36*n^3-12*n^2-6*n)*x+(36*n^5+36*n^4+9*n^3)*x^2)*P(n,x) = (64*n^3+64*n^2+16*n+(288*n^4+720*n^3+336*n^2-72*n-48)*x+(144*n^5+1224*n^4+1548*n^3-18*n^2-396*n-18)*x^2+(324*n^5+1080*n^4+837*n^3-135*n^2-135*n-27)*x^3)*P(n-1,x) - ((32*n^2+32*n+8)*x+(144*n^3+384*n^2+240*n+36)*x^2+(72*n^4+720*n^3+1206*n^2+558*n+54)*x^3+(216*n^4+864*n^3+1026*n^2+378*n+54)*x^4)*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^3 + 64*n^2 + 16*n + (288*n^4 + 720*n^3 + 336*n^2 - 72*n - 48)*x + (144*n^5 + 1224*n^4 + 1548*n^3 - 18*n^2 - 396*n - 18)*x^2 + (324*n^5 + 1080*n^4 + 837*n^3 - 135*n^2 - 135*n - 27)*x^3)*P[n - 1, x] - ((32*n^2 + 32*n + 8)*x + (144*n^3 + 384*n^2 + 240*n + 36)*x^2 + (72*n^4 + 720*n^3 + 1206*n^2 + 558*n + 54)*x^3 + (216*n^4 + 864*n^3 + 1026*n^2 + 378*n + 54)*x^4)*D[P[n - 1, x], x])/(16*n^3 + 16*n^2 + 4*n + (72*n^4 + 36*n^3 - 12*n^2 - 6*n)*x + (36*n^5 + 36*n^4 + 9*n^3)*x^2)]];\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^3 + 64*n^2 + 16*n + (288*n^4 + 720*n^3 + 336*n^2 - 72*n - 48)*x + (144*n^5 + 1224*n^4 + 1548*n^3 - 18*n^2 - 396*n - 18)*x^2 + (324*n^5 + 1080*n^4 + 837*n^3 - 135*n^2 - 135*n - 27)*x^3)*v[i - 1] - ((32*n^2 + 32*n + 8)*x + (144*n^3 + 384*n^2 + 240*n + 36)*x^2 + (72*n^4 + 720*n^3 + 1206*n^2 + 558*n + 54)*x^3 + (216*n^4 + 864*n^3 + 1026*n^2 + 378*n + 54)*x^4)*deriv(v[i - 1], x))/(16*n^3 + 16*n^2 + 4*n + (72*n^4 + 36*n^3 - 12*n^2 - 6*n)*x + (36*n^5 + 36*n^4 + 9*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2245b35ec7cd4857","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-0cc36dd83eba3acf","e-8a5dbf07cd515b88"],"text":"A091044 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A091044 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}