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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":"ac623575cb34a2709d31a9ef6c1b6c3e1600c9538d0cd82b6f11fb9f8535207a","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-2a15a23ed2e4e7bb","role":"sequence","name":"basis_map_A007318 of delannoy","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,     4;\n  2:    5,    20,      9;\n  3:   12,    72,     72,     16;\n  4:   29,   232,    360,    176,     25;\n  5:   70,   700,   1476,   1136,    350,     36;\n  6:  169,  2028,   5409,   5712,   2775,    612,    49;\n  7:  408,  5712,  18432,  24800,  16600,   5760,   980,    64;\n  8:  985, 15760,  59688,  97696,  83650,  40032, 10682,  1472,   81;\n  9: 2378, 42804, 186120, 359072, 375300, 230184, 84672, 18240, 2106, 100;\n","recurrence_sentence":"The row polynomials satisfy (32+(96*n-16)*x+(88*n^2-8*n-16)*x^2+(24*n^3+20*n^2-28*n-2)*x^3+(2*n^4+4*n^3-8*n^2-4*n+2)*x^4-(n^3+n^2)*x^5)*P(n,x) = (32+(96*n+112)*x+(88*n^2+264*n+144)*x^2+(24*n^3+180*n^2+252*n+78)*x^3+(2*n^4+36*n^3+108*n^2+84*n+10)*x^4+(2*n^4+11*n^3+12*n^2-5*n-6)*x^5-(n^3+4*n^2+5*n+2)*x^6)*P(n-2,x) + (64+(192*n+96)*x+(176*n^2+256*n+16)*x^2+(48*n^3+200*n^2+96*n-60)*x^3+(4*n^4+40*n^3+60*n^2-20*n-36)*x^4+(2*n^4+6*n^3-5*n^2-16*n-7)*x^5-(n^3+3*n^2+3*n+1)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((32 + (96*n + 112)*x + (88*n^2 + 264*n + 144)*x^2 + (24*n^3 + 180*n^2 + 252*n + 78)*x^3 + (2*n^4 + 36*n^3 + 108*n^2 + 84*n + 10)*x^4 + (2*n^4 + 11*n^3 + 12*n^2 - 5*n - 6)*x^5 - (n^3 + 4*n^2 + 5*n + 2)*x^6)*P[n - 2, x] + (64 + (192*n + 96)*x + (176*n^2 + 256*n + 16)*x^2 + (48*n^3 + 200*n^2 + 96*n - 60)*x^3 + (4*n^4 + 40*n^3 + 60*n^2 - 20*n - 36)*x^4 + (2*n^4 + 6*n^3 - 5*n^2 - 16*n - 7)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*P[n - 1, x])/(32 + (96*n - 16)*x + (88*n^2 - 8*n - 16)*x^2 + (24*n^3 + 20*n^2 - 28*n - 2)*x^3 + (2*n^4 + 4*n^3 - 8*n^2 - 4*n + 2)*x^4 - (n^3 + n^2)*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((32 + (96*n + 112)*x + (88*n^2 + 264*n + 144)*x^2 + (24*n^3 + 180*n^2 + 252*n + 78)*x^3 + (2*n^4 + 36*n^3 + 108*n^2 + 84*n + 10)*x^4 + (2*n^4 + 11*n^3 + 12*n^2 - 5*n - 6)*x^5 - (n^3 + 4*n^2 + 5*n + 2)*x^6)*v[i - 2] + (64 + (192*n + 96)*x + (176*n^2 + 256*n + 16)*x^2 + (48*n^3 + 200*n^2 + 96*n - 60)*x^3 + (4*n^4 + 40*n^3 + 60*n^2 - 20*n - 36)*x^4 + (2*n^4 + 6*n^3 - 5*n^2 - 16*n - 7)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*v[i - 1])/(32 + (96*n - 16)*x + (88*n^2 - 8*n - 16)*x^2 + (24*n^3 + 20*n^2 - 28*n - 2)*x^3 + (2*n^4 + 4*n^3 - 8*n^2 - 4*n + 2)*x^4 - (n^3 + n^2)*x^5)); 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-e00a1cb428776eaa","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-41d30650643fabff","e-5a8154d19ca698df"],"text":"delannoy \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"delannoy \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}