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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":"afada1aca54609d4590e412f25c9d3ed4a885bc2444c5c7f2eb56a14b5f76549","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fbf962df53148c52","role":"sequence","name":"basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.","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:           21,             40;\n  2:          441,           1680,           1200;\n  3:         9261,          52920,          75600,           32000;\n  4:       194481,        1481760,        3175200,         2688000,          800000;\n  5:      4084101,       38896200,      111132000,       141120000,        84000000,        19200000;\n  6:     85766121,      980184240,     3500658000,      5927040000,      5292000000,      2419200000,       448000000;\n  7:   1801088541,    24014513880,   102919345200,    217818720000,    259308000000,    177811200000,     65856000000,     10240000000;\n  8:  37822859361,   576348333120,  2881741665600,   7318708992000,  10890936000000,   9957427200000,   5531904000000,   1720320000000,   230400000000;\n  9: 794280046581, 13616229369960, 77807024971200, 230539333248000, 411677380800000, 470488435200000, 348509952000000, 162570240000000, 43545600000000, 5120000000000;\n","recurrence_sentence":"The row polynomials satisfy (21+20*n*x)*P(n,x) = (441+(420*n+840)*x+(400*n+400)*x^2)*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[((441 + (420*n + 840)*x + (400*n + 400)*x^2)*P[n - 1, x])/(21 + 20*n*x)]];\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] = ((441 + (420*n + 840)*x + (400*n + 400)*x^2)*v[i - 1])/(21 + 20*n*x)); 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-e1d1088f7d1ff894","steps":["basis A013609","\u03b8+1"],"edges":["e-25d25e50279a57da","e-5b91903b52c080c7"],"text":"A038324 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038324 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}