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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":"d8ec22c1427520d4d484b6c3e5523d2bb5148bf82a194b7e1c9cdc6afe8a51ee","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3897d3bccfbe11da","role":"sequence","name":"basis_map_A013609 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","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:     3,     16;\n  3:    10,     96,     108;\n  4:    35,    480,    1080,      512;\n  5:   126,   2240,    7560,     7680,     2000;\n  6:   462,  10080,   45360,    71680,    42000,     6912;\n  7:  1716,  44352,  249480,   537600,   504000,   193536,    21952;\n  8:  6435, 192192, 1297296,  3548160,  4620000,  2903040,   790272,   65536;\n  9: 24310, 823680, 6486480, 21525504, 36036000, 31933440, 14488320, 2949120, 186624;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4-8*n^3+5*n^2-n+(12*n^5-36*n^4+36*n^3-14*n^2+2*n)*x+(4*n^6-12*n^5+12*n^4-4*n^3)*x^2)*P(n,x) = (16*n^4-40*n^3+36*n^2-14*n+2+(48*n^5-72*n^4-40*n^3+120*n^2-68*n+12)*x+(16*n^6+64*n^5-220*n^4+136*n^3+88*n^2-108*n+24)*x^2+(24*n^6-24*n^5-48*n^4+72*n^3-40*n+16)*x^3)*P(n-1,x) - ((8*n^3-12*n^2+6*n-1)*x+(24*n^4-8*n^3-32*n^2+28*n-6)*x^2+(8*n^5+52*n^4-64*n^3-12*n^2+40*n-12)*x^3+(16*n^5+8*n^4-32*n^3+8*n^2+16*n-8)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^4 - 40*n^3 + 36*n^2 - 14*n + 2 + (48*n^5 - 72*n^4 - 40*n^3 + 120*n^2 - 68*n + 12)*x + (16*n^6 + 64*n^5 - 220*n^4 + 136*n^3 + 88*n^2 - 108*n + 24)*x^2 + (24*n^6 - 24*n^5 - 48*n^4 + 72*n^3 - 40*n + 16)*x^3)*P[n - 1, x] - ((8*n^3 - 12*n^2 + 6*n - 1)*x + (24*n^4 - 8*n^3 - 32*n^2 + 28*n - 6)*x^2 + (8*n^5 + 52*n^4 - 64*n^3 - 12*n^2 + 40*n - 12)*x^3 + (16*n^5 + 8*n^4 - 32*n^3 + 8*n^2 + 16*n - 8)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 8*n^3 + 5*n^2 - n + (12*n^5 - 36*n^4 + 36*n^3 - 14*n^2 + 2*n)*x + (4*n^6 - 12*n^5 + 12*n^4 - 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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^4 - 40*n^3 + 36*n^2 - 14*n + 2 + (48*n^5 - 72*n^4 - 40*n^3 + 120*n^2 - 68*n + 12)*x + (16*n^6 + 64*n^5 - 220*n^4 + 136*n^3 + 88*n^2 - 108*n + 24)*x^2 + (24*n^6 - 24*n^5 - 48*n^4 + 72*n^3 - 40*n + 16)*x^3)*v[i - 1] - ((8*n^3 - 12*n^2 + 6*n - 1)*x + (24*n^4 - 8*n^3 - 32*n^2 + 28*n - 6)*x^2 + (8*n^5 + 52*n^4 - 64*n^3 - 12*n^2 + 40*n - 12)*x^3 + (16*n^5 + 8*n^4 - 32*n^3 + 8*n^2 + 16*n - 8)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 8*n^3 + 5*n^2 - n + (12*n^5 - 36*n^4 + 36*n^3 - 14*n^2 + 2*n)*x + (4*n^6 - 12*n^5 + 12*n^4 - 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-b1b19325e0094118","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-169cfb9127a1a777","e-fbe6da501be6f0ce"],"text":"A367270 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A367270 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}