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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":"62e0b8637a146b20d6e5e0291e93efef553a0a9637d54880f41a6f9a407aecdb","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-f2968a7a3d3a43cd","role":"sequence","name":"veronese_section_2_0 of Triangle of coefficients in expansion of (2+3x)^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:   2;\n  2:   4,     36;\n  3:   8,    216;\n  4:  16,    864,     729;\n  5:  32,   2880,    7290;\n  6:  64,   8640,   43740,   11664;\n  7: 128,  24192,  204120,  163296;\n  8: 256,  64512,  816480, 1306368,  164025;\n  9: 512, 165888, 2939328, 7838208, 2952450;\n","recurrence_sentence":"The row polynomials satisfy (128*n-128-(176*n^3-192*n^2-80*n+96)*x+(18*n^5+18*n^4-18*n^3-18*n^2)*x^2)*P(n,x) = (256*n-256-(352*n^3-384*n^2-2464*n+2496)*x+(36*n^5+36*n^4-1404*n^3-1044*n^2+2232*n+144)*x^2+(81*n^5+405*n^4+567*n^3-81*n^2-648*n-324)*x^3)*P(n-1,x) + (512*x-(704*n^2+1344*n+1408)*x^2+(72*n^4+432*n^3+2520*n^2+3888*n+864)*x^3-(162*n^4+972*n^3+2106*n^2+1944*n+648)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((256*n - 256 - (352*n^3 - 384*n^2 - 2464*n + 2496)*x + (36*n^5 + 36*n^4 - 1404*n^3 - 1044*n^2 + 2232*n + 144)*x^2 + (81*n^5 + 405*n^4 + 567*n^3 - 81*n^2 - 648*n - 324)*x^3)*P[n - 1, x] + (512*x - (704*n^2 + 1344*n + 1408)*x^2 + (72*n^4 + 432*n^3 + 2520*n^2 + 3888*n + 864)*x^3 - (162*n^4 + 972*n^3 + 2106*n^2 + 1944*n + 648)*x^4)*D[P[n - 1, x], x])/(128*n - 128 - (176*n^3 - 192*n^2 - 80*n + 96)*x + (18*n^5 + 18*n^4 - 18*n^3 - 18*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; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((256*n - 256 - (352*n^3 - 384*n^2 - 2464*n + 2496)*x + (36*n^5 + 36*n^4 - 1404*n^3 - 1044*n^2 + 2232*n + 144)*x^2 + (81*n^5 + 405*n^4 + 567*n^3 - 81*n^2 - 648*n - 324)*x^3)*v[i - 1] + (512*x - (704*n^2 + 1344*n + 1408)*x^2 + (72*n^4 + 432*n^3 + 2520*n^2 + 3888*n + 864)*x^3 - (162*n^4 + 972*n^3 + 2106*n^2 + 1944*n + 648)*x^4)*deriv(v[i - 1], x))/(128*n - 128 - (176*n^3 - 192*n^2 - 80*n + 96)*x + (18*n^5 + 18*n^4 - 18*n^3 - 18*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-5733d79f45a14084","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-bdf59334244a08a5","e-ae357d24a7e55748"],"text":"A013620 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A013620 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}