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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":"952df9e9e7e8361cfa13abf1d760e2dd15c71f7912dc9a7431a4b7cd7f89d73e","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-53dc6d1eeb1907f4","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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:            324,             2016,              1764;\n  1:           5832,            54432,             95256,             43904;\n  2:         104976,          1306368,           3429216,           3161088,             960400;\n  3:        1889568,         29393280,         102876480,         142248960,           86436000,           19361664;\n  4:       34012224,        634894848,        2777664960,        5120962560,         4667544000,         2091059712,          368947264;\n  5:      612220032,      13332791808,       69997156992,      161310320640,       196036848000,       131736761856,        46487355264,         6746464256;\n  6:    11019960576,     274274574336,     1679931767808,     4645737234432,      7057326528000,      6323364569088,      3347089579008,       971490852864,       119538913536;\n  7:   198359290368,    5554060130304,    38878420912128,   125434905329664,    228657379507200,    256096265048064,    180742837266432,     78690759081984,     19365303992832,     2066104678400;\n  8:  3570467226624,  111081202606080,   874764470522880,  3225468994191360,   6859721385216000,   9219465541730304,   8133427676989440,   4721445544919040,   1742877359354880,   371898842112000,   34999813252096;\n  9: 64268410079232, 2199407811600384, 19244818351503360, 79830357606236160, 194032119181824000, 304242362877100032, 322083736008781824, 233711554473492480, 115029905717422080, 36817985369088000, 6929963023915008, 583137384431616;\n","recurrence_sentence":"The row polynomials satisfy (81+(189*n+315)*x+(49*n^2+196*n+196)*x^2)*P(n,x) = (1458+(3402*n+10206)*x+(882*n^2+7938*n+14994)*x^2+(686*n^2+4116*n+6174)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 2016*x + 1764*x^2.","mathematica":"P[0, x_] := 324 + 2016*x + 1764*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((1458 + (3402*n + 10206)*x + (882*n^2 + 7938*n + 14994)*x^2 + (686*n^2 + 4116*n + 6174)*x^3)*P[n - 1, x])/(81 + (189*n + 315)*x + (49*n^2 + 196*n + 196)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 324 + 2016*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1458 + (3402*n + 10206)*x + (882*n^2 + 7938*n + 14994)*x^2 + (686*n^2 + 4116*n + 6174)*x^3)*v[i - 1])/(81 + (189*n + 315)*x + (49*n^2 + 196*n + 196)*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-7dcf11eddf2b9a8a","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-322a2e4c0bbd3231","e-9732e6c454fc8a66"],"text":"A038321 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038321 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}