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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":"96b80243fb104b6a77b750aad6feb615d8480328e45ea1b17a413210d00e3925","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-b2f14d9f44a39df2","role":"sequence","name":"Triangle of coefficients of Legendre polynomials 2^n P_n (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;\n  2:     6,      8;\n  3:    20,     48;\n  4:    70,    240,     54;\n  5:   252,   1120,    540;\n  6:   924,   5040,   3780,    320;\n  7:  3432,  22176,  22680,   4480;\n  8: 12870,  96096, 124740,  40320,  1750;\n  9: 48620, 411840, 648648, 295680, 31500;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3+16*n^2+4*n+(12*n^4+6*n^3-6*n^2-4*n)*x+(n^5+2*n^4+n^3)*x^2)*P(n,x) = (64*n^3+32*n^2-16*n-8+(48*n^4+128*n^3-36*n^2-100*n-24)*x+(4*n^5+46*n^4+52*n^3-34*n^2-36*n-24)*x^2+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^3)*P(n-1,x) - ((64*n^2+64*n+16)*x+(48*n^3+184*n^2+152*n+48)*x^2+(4*n^4+72*n^3+172*n^2+136*n+48)*x^3+(4*n^4+24*n^3+52*n^2+48*n+16)*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 + 32*n^2 - 16*n - 8 + (48*n^4 + 128*n^3 - 36*n^2 - 100*n - 24)*x + (4*n^5 + 46*n^4 + 52*n^3 - 34*n^2 - 36*n - 24)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*P[n - 1, x] - ((64*n^2 + 64*n + 16)*x + (48*n^3 + 184*n^2 + 152*n + 48)*x^2 + (4*n^4 + 72*n^3 + 172*n^2 + 136*n + 48)*x^3 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*D[P[n - 1, x], x])/(16*n^3 + 16*n^2 + 4*n + (12*n^4 + 6*n^3 - 6*n^2 - 4*n)*x + (n^5 + 2*n^4 + 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 + 32*n^2 - 16*n - 8 + (48*n^4 + 128*n^3 - 36*n^2 - 100*n - 24)*x + (4*n^5 + 46*n^4 + 52*n^3 - 34*n^2 - 36*n - 24)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*v[i - 1] - ((64*n^2 + 64*n + 16)*x + (48*n^3 + 184*n^2 + 152*n + 48)*x^2 + (4*n^4 + 72*n^3 + 172*n^2 + 136*n + 48)*x^3 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*deriv(v[i - 1], x))/(16*n^3 + 16*n^2 + 4*n + (12*n^4 + 6*n^3 - 6*n^2 - 4*n)*x + (n^5 + 2*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b2f14d9f44a39df2","steps":["(\u03b8+1)\u00b2"],"edges":["e-5c14ddbea863bc35"],"text":"A008556 \u2192 (\u03b8+1)\u00b2"},"headline":"A008556 \u2192 (\u03b8+1)\u00b2","chain":null}