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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":"2c8239bdd9fdd0012e4d023ead15e9dc6c16f3238729ffb66debfd117b86310f","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-90af30f849fc554d","role":"sequence","name":"Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 8.","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: 1,         4;\n  2: 1,        72,            9;\n  3: 1,       716,         1611,            16;\n  4: 1,      6544,        54774,         26176,             25;\n  5: 1,     59028,      1299258,       2309792,         368925,             36;\n  6: 1,    531416,     26470215,     115489600,       73528375,        4782744,             49;\n  7: 1,   4782940,    499016781,    4440159280,     6937748875,     1996067124,       58591015,            64;\n  8: 1,  43046688,   9010604412,  147376760704,   457891425750,   331597711584,    49057735132,     688747008,         81;\n  9: 1, 387420452, 158701012740, 4469342779712, 24551740782350, 35354506726584, 13687362262868, 1128540535040, 7845264153, 100;\n","recurrence_sentence":"The row polynomials satisfy (64*n^2-112*n+49)*P(n,x) = (448*n^2-336*n-49-(2048*n^3-5376*n^2-11680*n+27300)*x+(36864*n^4-137216*n^3+162176*n^2-67424*n+5145)*x^2)*P(n-2,x) - ((1024*n^2-768*n-4144)*x-(2048*n^3+13056*n^2-7072*n-23100)*x^2-(16384*n^4-114688*n^3+172544*n^2-82432*n+7252)*x^3)*d/dx P(n-2,x) - ((4096*n^2-3072*n-4032)*x^2-(16384*n^3-26624*n^2+13056*n-5600)*x^3+(16384*n^3-30720*n^2+16128*n-1568)*x^4)*d^2/dx^2 P(n-2,x) - (384*n^2-224*n-98-(2048*n^3-5376*n^2+4704*n-1372)*x)*P(n-1,x) + ((1024*n^2-1280*n-112)*x-(2048*n^3-3328*n^2+1120*n+196)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((448*n^2 - 336*n - 49 - (2048*n^3 - 5376*n^2 - 11680*n + 27300)*x + (36864*n^4 - 137216*n^3 + 162176*n^2 - 67424*n + 5145)*x^2)*P[n - 2, x] - ((1024*n^2 - 768*n - 4144)*x - (2048*n^3 + 13056*n^2 - 7072*n - 23100)*x^2 - (16384*n^4 - 114688*n^3 + 172544*n^2 - 82432*n + 7252)*x^3)*D[P[n - 2, x], x] - ((4096*n^2 - 3072*n - 4032)*x^2 - (16384*n^3 - 26624*n^2 + 13056*n - 5600)*x^3 + (16384*n^3 - 30720*n^2 + 16128*n - 1568)*x^4)*D[P[n - 2, x], {x, 2}] - (384*n^2 - 224*n - 98 - (2048*n^3 - 5376*n^2 + 4704*n - 1372)*x)*P[n - 1, x] + ((1024*n^2 - 1280*n - 112)*x - (2048*n^3 - 3328*n^2 + 1120*n + 196)*x^2)*D[P[n - 1, x], x])/(64*n^2 - 112*n + 49)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((448*n^2 - 336*n - 49 - (2048*n^3 - 5376*n^2 - 11680*n + 27300)*x + (36864*n^4 - 137216*n^3 + 162176*n^2 - 67424*n + 5145)*x^2)*v[i - 2] - ((1024*n^2 - 768*n - 4144)*x - (2048*n^3 + 13056*n^2 - 7072*n - 23100)*x^2 - (16384*n^4 - 114688*n^3 + 172544*n^2 - 82432*n + 7252)*x^3)*deriv(v[i - 2], x) - ((4096*n^2 - 3072*n - 4032)*x^2 - (16384*n^3 - 26624*n^2 + 13056*n - 5600)*x^3 + (16384*n^3 - 30720*n^2 + 16128*n - 1568)*x^4)*deriv(deriv(v[i - 2], x), x) - (384*n^2 - 224*n - 98 - (2048*n^3 - 5376*n^2 + 4704*n - 1372)*x)*v[i - 1] + ((1024*n^2 - 1280*n - 112)*x - (2048*n^3 - 3328*n^2 + 1120*n + 196)*x^2)*deriv(v[i - 1], x))/(64*n^2 - 112*n + 49)); 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-90af30f849fc554d","steps":["(\u03b8+1)\u00b2"],"edges":["e-9cd2941aaa72a6a2"],"text":"A167884 \u2192 (\u03b8+1)\u00b2"},"headline":"A167884 \u2192 (\u03b8+1)\u00b2","chain":null}