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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":"ca04e875163acfe39751761799480dc93d2c44280093f616e251fb1aee9374de","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-a1e0668357b9b581","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).","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,        16;\n  2:      35,       192,        144;\n  3:     204,      1696,       2592,       1024;\n  4:    1189,     13248,      30672,      24576,       6400;\n  5:    6930,     96816,     300672,     364544,     192000,      36864;\n  6:   40391,    678528,    2644704,    4300800,    3424000,    1327104,     200704;\n  7:  235416,   4620608,   21674304,   44247040,   47232000,   27648000,    8429568,    1048576;\n  8: 1372105,  30811776,  168986592,  415309824,  556512000,  436617216,  200904704,   50331648,   5308416;\n  9: 7997214, 202203728, 1269551232, 3649294336, 5887526400, 5797564416, 3574136832, 1350565888, 286654464, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (64*n+12*n*x-(80*n^3-9*n)*x^2-30*n^3*x^3+4*n^5*x^4)*P(n,x) = (192*n+192+(224*n^2+548*n+324)*x-(240*n^3+264*n^2-123*n-147)*x^2-(32*n^4+362*n^3+602*n^2+296*n+24)*x^3+(12*n^5+12*n^4-108*n^3-216*n^2-132*n-24)*x^4+(8*n^5+24*n^4+24*n^3+8*n^2)*x^5)*P(n-1,x) + (64+144*x-(224*n^2+224*n-76)*x^2-(456*n^2+456*n+42)*x^3+(16*n^4+32*n^3-276*n^2-292*n-50)*x^4+(24*n^4+48*n^3-36*n^2-60*n-12)*x^5+(8*n^4+16*n^3+8*n^2)*x^6)*d/dx 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[((192*n + 192 + (224*n^2 + 548*n + 324)*x - (240*n^3 + 264*n^2 - 123*n - 147)*x^2 - (32*n^4 + 362*n^3 + 602*n^2 + 296*n + 24)*x^3 + (12*n^5 + 12*n^4 - 108*n^3 - 216*n^2 - 132*n - 24)*x^4 + (8*n^5 + 24*n^4 + 24*n^3 + 8*n^2)*x^5)*P[n - 1, x] + (64 + 144*x - (224*n^2 + 224*n - 76)*x^2 - (456*n^2 + 456*n + 42)*x^3 + (16*n^4 + 32*n^3 - 276*n^2 - 292*n - 50)*x^4 + (24*n^4 + 48*n^3 - 36*n^2 - 60*n - 12)*x^5 + (8*n^4 + 16*n^3 + 8*n^2)*x^6)*D[P[n - 1, x], x])/(64*n + 12*n*x - (80*n^3 - 9*n)*x^2 - 30*n^3*x^3 + 4*n^5*x^4)]];\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] = ((192*n + 192 + (224*n^2 + 548*n + 324)*x - (240*n^3 + 264*n^2 - 123*n - 147)*x^2 - (32*n^4 + 362*n^3 + 602*n^2 + 296*n + 24)*x^3 + (12*n^5 + 12*n^4 - 108*n^3 - 216*n^2 - 132*n - 24)*x^4 + (8*n^5 + 24*n^4 + 24*n^3 + 8*n^2)*x^5)*v[i - 1] + (64 + 144*x - (224*n^2 + 224*n - 76)*x^2 - (456*n^2 + 456*n + 42)*x^3 + (16*n^4 + 32*n^3 - 276*n^2 - 292*n - 50)*x^4 + (24*n^4 + 48*n^3 - 36*n^2 - 60*n - 12)*x^5 + (8*n^4 + 16*n^3 + 8*n^2)*x^6)*deriv(v[i - 1], x))/(64*n + 12*n*x - (80*n^3 - 9*n)*x^2 - 30*n^3*x^3 + 4*n^5*x^4)); 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-c2b7c74f8491f01b","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-3938f5b848156b83","e-f1d7611539a68465"],"text":"A207824 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A207824 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}