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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":"1ee39d12d49a3656349295a89650a7ac58443f91fd993877d5e397b2daee6de6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6a6677e3386580cd","role":"sequence","name":"basis_map_A013609 of Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).","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:    1;\n  2:    2,     8;\n  3:    4,    24;\n  4:   10,    96,    108;\n  5:   26,   320,    540;\n  6:   76,  1200,   3240,   1920;\n  7:  232,  4368,  15120,  13440;\n  8:  764, 17024,  75600, 107520,  42000;\n  9: 2620, 66816, 353808, 645120, 378000;\n","recurrence_sentence":"The row polynomials satisfy (4+(10*n+12)*x+(14*n^2+10*n+8)*x^2+(6*n^3+14*n^2-4*n-8)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (4*n+(17*n^2+23*n-18)*x+(20*n^3+76*n^2+24*n-88)*x^2+(8*n^4+72*n^3+88*n^2-48*n-112)*x^3+(n^5+22*n^4+65*n^3+12*n^2-80*n-16)*x^4+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) - (1+(6*n+12)*x+(11*n^2+49*n+52)*x^2+(6*n^3+70*n^2+156*n+104)*x^3+(n^4+30*n^3+161*n^2+240*n+100)*x^4+(4*n^4+48*n^3+156*n^2+176*n+48)*x^5+(4*n^4+24*n^3+52*n^2+48*n+16)*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[((4*n + (17*n^2 + 23*n - 18)*x + (20*n^3 + 76*n^2 + 24*n - 88)*x^2 + (8*n^4 + 72*n^3 + 88*n^2 - 48*n - 112)*x^3 + (n^5 + 22*n^4 + 65*n^3 + 12*n^2 - 80*n - 16)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] - (1 + (6*n + 12)*x + (11*n^2 + 49*n + 52)*x^2 + (6*n^3 + 70*n^2 + 156*n + 104)*x^3 + (n^4 + 30*n^3 + 161*n^2 + 240*n + 100)*x^4 + (4*n^4 + 48*n^3 + 156*n^2 + 176*n + 48)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(4 + (10*n + 12)*x + (14*n^2 + 10*n + 8)*x^2 + (6*n^3 + 14*n^2 - 4*n - 8)*x^3 + (n^4 + 2*n^3 + 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] = ((4*n + (17*n^2 + 23*n - 18)*x + (20*n^3 + 76*n^2 + 24*n - 88)*x^2 + (8*n^4 + 72*n^3 + 88*n^2 - 48*n - 112)*x^3 + (n^5 + 22*n^4 + 65*n^3 + 12*n^2 - 80*n - 16)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] - (1 + (6*n + 12)*x + (11*n^2 + 49*n + 52)*x^2 + (6*n^3 + 70*n^2 + 156*n + 104)*x^3 + (n^4 + 30*n^3 + 161*n^2 + 240*n + 100)*x^4 + (4*n^4 + 48*n^3 + 156*n^2 + 176*n + 48)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(4 + (10*n + 12)*x + (14*n^2 + 10*n + 8)*x^2 + (6*n^3 + 14*n^2 - 4*n - 8)*x^3 + (n^4 + 2*n^3 + 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-97b51734d39ede6c","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-4139d19ff9e22325","e-144c06176639f69e"],"text":"A100861 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A100861 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}