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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":"06ddf08a66956adcf352f95d78d7543efb97d5317c23ccfea69cd3ae5b554421","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-92efd0b8c392ac45","role":"sequence","name":"gamma_polynomial of Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.","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:  3;\n  2:  6,   12;\n  3: 10,   60;\n  4: 15,  180,    81;\n  5: 21,  420,   567;\n  6: 28,  840,  2268,   448;\n  7: 36, 1512,  6804,  4032;\n  8: 45, 2520, 17010, 20160,  2250;\n  9: 55, 3960, 37422, 73920, 24750;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+24*n+(5*n^4+27*n^3-22*n^2-24*n)*x+(n^6+8*n^5+13*n^4+6*n^3)*x^2)*P(n,x) = (4*n^2+32*n+48+(5*n^4+37*n^3+96*n^2-4*n-176)*x+(n^6+10*n^5+61*n^4+96*n^3-52*n^2-96*n+64)*x^2+(4*n^6+28*n^5+68*n^4+52*n^3-40*n^2-80*n-32)*x^3)*P(n-1,x) + ((8*n+16)*x+(10*n^3+34*n^2-8*n-72)*x^2+(2*n^5+16*n^4+10*n^3-60*n^2-40*n+48)*x^3-(8*n^5+64*n^4+200*n^3+304*n^2+224*n+64)*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[((4*n^2 + 32*n + 48 + (5*n^4 + 37*n^3 + 96*n^2 - 4*n - 176)*x + (n^6 + 10*n^5 + 61*n^4 + 96*n^3 - 52*n^2 - 96*n + 64)*x^2 + (4*n^6 + 28*n^5 + 68*n^4 + 52*n^3 - 40*n^2 - 80*n - 32)*x^3)*P[n - 1, x] + ((8*n + 16)*x + (10*n^3 + 34*n^2 - 8*n - 72)*x^2 + (2*n^5 + 16*n^4 + 10*n^3 - 60*n^2 - 40*n + 48)*x^3 - (8*n^5 + 64*n^4 + 200*n^3 + 304*n^2 + 224*n + 64)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 24*n + (5*n^4 + 27*n^3 - 22*n^2 - 24*n)*x + (n^6 + 8*n^5 + 13*n^4 + 6*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] = ((4*n^2 + 32*n + 48 + (5*n^4 + 37*n^3 + 96*n^2 - 4*n - 176)*x + (n^6 + 10*n^5 + 61*n^4 + 96*n^3 - 52*n^2 - 96*n + 64)*x^2 + (4*n^6 + 28*n^5 + 68*n^4 + 52*n^3 - 40*n^2 - 80*n - 32)*x^3)*v[i - 1] + ((8*n + 16)*x + (10*n^3 + 34*n^2 - 8*n - 72)*x^2 + (2*n^5 + 16*n^4 + 10*n^3 - 60*n^2 - 40*n + 48)*x^3 - (8*n^5 + 64*n^4 + 200*n^3 + 304*n^2 + 224*n + 64)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 24*n + (5*n^4 + 27*n^3 - 22*n^2 - 24*n)*x + (n^6 + 8*n^5 + 13*n^4 + 6*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-bb74f03367de3ec8","steps":["\u03b3","(\u03b8+1)\u00b2"],"edges":["e-fbbe2b618491014f","e-c1011a806d9ddf62"],"text":"A145597 \u2192 \u03b3 \u2192 (\u03b8+1)\u00b2"},"headline":"A145597 \u2192 \u03b3 \u2192 (\u03b8+1)\u00b2","chain":null}