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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":"bf94a15da76dec57d4c005b12dc8c2c817965a0b0ec9a5ea30bd293d728d5522","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-66d8297eef1ced69","role":"sequence","name":"basis_map_A007318 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,            900;\n  3:         9261,          52920,          56700,           16000;\n  4:       194481,        1481760,        2381400,         1344000,          250000;\n  5:      4084101,       38896200,       83349000,        70560000,        26250000,        3600000;\n  6:     85766121,      980184240,     2625493500,      2963520000,      1653750000,      453600000,       49000000;\n  7:   1801088541,    24014513880,    77189508900,    108909360000,     81033750000,    33339600000,     7203000000,      640000000;\n  8:  37822859361,   576348333120,  2161306249200,   3659354496000,   3403417500000,  1867017600000,   605052000000,   107520000000,    8100000000;\n  9: 794280046581, 13616229369960, 58355268728400, 115269666624000, 128649181500000, 88216581600000, 38118276000000, 10160640000000, 1530900000000, 100000000000;\n","recurrence_sentence":"The row polynomials satisfy (441+(630*n-210)*x+100*n^2*x^2)*P(n,x) = (9261+(13230*n+13230)*x+(2100*n^2+10500*n+6300)*x^2+(1000*n^2+2000*n+1000)*x^3)*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[((9261 + (13230*n + 13230)*x + (2100*n^2 + 10500*n + 6300)*x^2 + (1000*n^2 + 2000*n + 1000)*x^3)*P[n - 1, x])/(441 + (630*n - 210)*x + 100*n^2*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] = ((9261 + (13230*n + 13230)*x + (2100*n^2 + 10500*n + 6300)*x^2 + (1000*n^2 + 2000*n + 1000)*x^3)*v[i - 1])/(441 + (630*n - 210)*x + 100*n^2*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-e1d1088f7d1ff894","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-3706f9e3ae504ed3","e-0b682ef41885a51a"],"text":"A038324 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A038324 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}