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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":"90ffddb19c5d0e6d39b25a63b8d9548399840f8648ca026679e3259f9b84a740","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-e03395d80bc5749d","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^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:         11,          16;\n  2:        121,         352,          144;\n  3:       1331,        5808,         4752,         1024;\n  4:      14641,       85184,       104544,        45056,         6400;\n  5:     161051,     1171280,      1916640,      1239040,       352000,       36864;\n  6:    1771561,    15460896,     31624560,     27258880,     11616000,     2433024,      200704;\n  7:   19487171,   198414832,    487018224,    524733440,    298144000,    93671424,    15454208,    1048576;\n  8:  214358881,  2494357888,   7142933952,   9235308544,   6559168000,  2747695104,   679985152,   92274688,   5308416;\n  9: 2357947691, 30867678864, 101021494464, 152382590976, 129871526400, 68005453824, 22439510016, 4567597056, 525533184, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (121+(132*n-44)*x+16*n^2*x^2)*P(n,x) = (1331+(1452*n+1452)*x+(176*n^2+880*n+528)*x^2+(64*n^2+128*n+64)*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[((1331 + (1452*n + 1452)*x + (176*n^2 + 880*n + 528)*x^2 + (64*n^2 + 128*n + 64)*x^3)*P[n - 1, x])/(121 + (132*n - 44)*x + 16*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] = ((1331 + (1452*n + 1452)*x + (176*n^2 + 880*n + 528)*x^2 + (64*n^2 + 128*n + 64)*x^3)*v[i - 1])/(121 + (132*n - 44)*x + 16*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-a920b56c9b3ef4d1","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-551d764375152d3c","e-03abf34bd7b4dca4"],"text":"A038303 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A038303 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}