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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":"44197965c073d5c0ffbeb8d1da08bd68d93178c4c63dc910dfe91551fcfc7fff","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-117b55f7953e214b","role":"sequence","name":"basis_map_A013610 of Square the entries of Pascal's triangle.","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,      12;\n  2:     6,      72,       81;\n  3:    20,     360,      972,       432;\n  4:    70,    1680,     7290,      8640,      2025;\n  5:   252,    7560,    45360,     90720,     60750,      8748;\n  6:   924,   33264,   255150,    725760,    850500,    367416,     35721;\n  7:  3432,  144144,  1347192,   4989600,   8505000,   6613488,   2000376,    139968;\n  8: 12870,  617760,  6810804,  31135104,  70166250,  80831520,  45008460,  10077696,   531441;\n  9: 48620, 2625480, 33359040, 181621440, 510810300, 788107320, 660124080, 277136640, 47829690, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+(18*n^3-9*n^2)*x+9*n^4*x^2)*P(n,x) = (16*n^2-8*n+(72*n^3+72*n^2-78*n)*x+(36*n^4+252*n^3+18*n^2-198*n+18)*x^2+(81*n^4+162*n^3-27*n^2-108*n)*x^3)*P(n-1,x) - (8*n*x+(36*n^2+60*n-3)*x^2+(18*n^3+162*n^2+144*n-9)*x^3+(54*n^3+162*n^2+108*n)*x^4)*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[((16*n^2 - 8*n + (72*n^3 + 72*n^2 - 78*n)*x + (36*n^4 + 252*n^3 + 18*n^2 - 198*n + 18)*x^2 + (81*n^4 + 162*n^3 - 27*n^2 - 108*n)*x^3)*P[n - 1, x] - (8*n*x + (36*n^2 + 60*n - 3)*x^2 + (18*n^3 + 162*n^2 + 144*n - 9)*x^3 + (54*n^3 + 162*n^2 + 108*n)*x^4)*D[P[n - 1, x], x])/(4*n^2 + (18*n^3 - 9*n^2)*x + 9*n^4*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] = ((16*n^2 - 8*n + (72*n^3 + 72*n^2 - 78*n)*x + (36*n^4 + 252*n^3 + 18*n^2 - 198*n + 18)*x^2 + (81*n^4 + 162*n^3 - 27*n^2 - 108*n)*x^3)*v[i - 1] - (8*n*x + (36*n^2 + 60*n - 3)*x^2 + (18*n^3 + 162*n^2 + 144*n - 9)*x^3 + (54*n^3 + 162*n^2 + 108*n)*x^4)*deriv(v[i - 1], x))/(4*n^2 + (18*n^3 - 9*n^2)*x + 9*n^4*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-b13531ad79e7c629","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-0e684cdba462224e","e-f09ad7ac7aee8c3f"],"text":"A008459 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A008459 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}