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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":"3201826ec0959858febaed5a063a85d49d74d7ab7b718018a176de863945d290","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-586959989bac3453","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^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:             400,             3520,               5808;\n  1:            8000,           105600,             348480,              340736;\n  2:          160000,          2816000,           13939200,            27258880,            18740480;\n  3:         3200000,         70400000,          464640000,          1362944000,          1874048000,            989497344;\n  4:        64000000,       1689600000,        13939200000,         54517760000,        112442880000,         118739681280,          50794196992;\n  5:      1280000000,      39424000000,       390297600000,       1908121600000,       5247334400000,        8311777689600,        7111187578880,         2554222477312;\n  6:     25600000000,     901120000000,     10407936000000,      61059891200000,     209893376000000,      443294810112000,      568895006310400,       408675596369920,       126434012626944;\n  7:    512000000000,   20275200000000,    267632640000000,    1831796736000000,    7556161536000000,    19948266455040000,    34133700378624000,     36780803673292800,     22758122272849920,      6181218395095040;\n  8:  10240000000000,  450560000000000,   6690816000000000,   52337049600000000,  251872051200000000,   797930658201600000,  1706685018931200000,   2452053578219520000,   2275812227284992000,   1236243679019008000,   299170970322599936;\n  9: 204800000000000, 9912320000000000, 163553280000000000, 1439268864000000000, 7915978752000000000, 29257457467392000000, 75094140832972800000, 134862946802073600000, 166892896667566080000, 135986804692090880000, 65817613470971985920, 14360206575484796928;\n","recurrence_sentence":"The row polynomials satisfy (5+(11*n+22)*x)*P(n,x) = (100+(220*n+880)*x+(484*n+1452)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 3520*x + 5808*x^2.","mathematica":"P[0, x_] := 400 + 3520*x + 5808*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((100 + (220*n + 880)*x + (484*n + 1452)*x^2)*P[n - 1, x])/(5 + (11*n + 22)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 400 + 3520*x + 5808*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((100 + (220*n + 880)*x + (484*n + 1452)*x^2)*v[i - 1])/(5 + (11*n + 22)*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-cfc5b48d32898a3e","steps":["basis A013611","\u03b8+1"],"edges":["e-9ac6ce8d07a161c9","e-1600b593bf39ef5e"],"text":"A038301 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038301 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}