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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":"6a3e141db2079b6594066772da1c5400603a2d658a577cbc4933bad6b2ccf419","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-2ff1c5d05cc7026f","role":"sequence","name":"basis_map_A013610 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^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:        6,        42,          54;\n  2:       36,       504,        1971,         3024,          1620;\n  3:      216,      4536,       29646,        91476,        148230,        122472,          40824;\n  4:     1296,     36288,      332424,      1542240,       4180005,       6940080,        6980904,         3919104,          944784;\n  5:     7776,    272160,     3207600,     19867680,      75536550,     188617086,      317253510,       357618240,       259815600,       110224800,         20785248;\n  6:    46656,   1959552,    28238544,    218816640,    1070973900,    3571650936,     8428160727,     14286603744,     17349777180,     14770123200,       8386847568,       2857026816,        442158912;\n  7:   279936,  13716864,   233676576,   2174122944,   13016936520,   54326497176,   165062954574,    373715102376,    636671396214,    814897457640,     773206029288,     528311875392,     246061434528,      69997156992,       9183300480;\n  8:  1679616,  94058496,  1849257216,  20081488896,  142182643680,  712687102848,  2649403708560,   7507637456256,  16469076048201,  28153640460960,   37470138163920,   38485103553792,   29943664759008,   17079306306048,    6740542552320,   1645647446016,    187339329792;\n  9: 10077696, 634894848, 14149085184, 175654241280, 1435712662080, 8401325438016, 36940258891872, 125736004211328, 337497162734214, 722122126656210, 1237489596692118, 1697436056852928, 1852290124435296, 1587850507785024, 1046634530656320, 512207767572480, 175349612685312, 37489905879552, 3768826516992;\n","recurrence_sentence":"The row polynomials satisfy (2+7*n*x+(12*n-6)*x^2)*P(n,x) = (12+(42*n+84)*x+(219*n+219)*x^2+(378*n+252)*x^3+(216*n+108)*x^4)*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[((12 + (42*n + 84)*x + (219*n + 219)*x^2 + (378*n + 252)*x^3 + (216*n + 108)*x^4)*P[n - 1, x])/(2 + 7*n*x + (12*n - 6)*x^2)]];\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] = ((12 + (42*n + 84)*x + (219*n + 219)*x^2 + (378*n + 252)*x^3 + (216*n + 108)*x^4)*v[i - 1])/(2 + 7*n*x + (12*n - 6)*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-eed4f744ce063213","steps":["basis A013610","\u03b8+1"],"edges":["e-bf76d3591eaf645b","e-f831077fc9ef864c"],"text":"A200536 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A200536 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}