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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":"a3d77ef3423671cc8335503c30c7116cc7184522a056ca8271f504960fb1667c","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-a6cd3f3657ee08fd","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (3+5x)^n.","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:         8,         30;\n  2:        64,        480,         675;\n  3:       512,       5760,       16200,        13500;\n  4:      4096,      61440,      259200,       432000,        253125;\n  5:     32768,     614400,     3456000,      8640000,      10125000,       4556250;\n  6:    262144,    5898240,    41472000,    138240000,     243000000,     218700000,      79734375;\n  7:   2097152,   55050240,   464486400,   1935360000,    4536000000,    6123600000,    4465125000,    1366875000;\n  8:  16777216,  503316480,  4954521600,  24772608000,   72576000000,  130636800000,  142884000000,   87480000000,   23066015625;\n  9: 134217728, 4529848320, 50960793600, 297271296000, 1045094400000, 2351462400000, 3429216000000, 3149280000000, 1660753125000, 384433593750;\n","recurrence_sentence":"The row polynomials satisfy (8+15*n*x)*P(n,x) = (64+(120*n+240)*x+(225*n+225)*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[((64 + (120*n + 240)*x + (225*n + 225)*x^2)*P[n - 1, x])/(8 + 15*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] = ((64 + (120*n + 240)*x + (225*n + 225)*x^2)*v[i - 1])/(8 + 15*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-c44f7114310405ad","steps":["basis A013610","\u03b8+1"],"edges":["e-c78a98c6ca91439c","e-658aebc2fbe07cf8"],"text":"A013622 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A013622 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}