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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":"56a6c572f4d21089b000368f2b43882e959059777a9e04e1169621867dd8f70d","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-1453c6ad660cd835","role":"sequence","name":"reversal of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","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:  2,    6,    3;\n  1:  4,   16,   15,     4;\n  2:  4,   24,   39,    24,     5;\n  3:  8,   56,  114,   100,    40,     6;\n  4:  8,   72,  198,   252,   165,    54,     7;\n  5: 16,  160,  504,   768,   645,   306,    77,     8;\n  6: 16,  192,  744,  1440,  1605,  1080,   434,    96,    9;\n  7: 32,  416, 1776,  3872,  5010,  4086,  2128,   688,  126,   10;\n  8: 32,  480, 2400,  6240,  9850, 10098,  6895,  3120,  900,  150,  11;\n  9: 64, 1024, 5520, 15680, 27500, 32016, 25571, 14120, 5310, 1300, 187, 12;\n","recurrence_sentence":"The row polynomials satisfy (32+(24*n+92)*x+(9*n^2+59*n+96)*x^2+(12*n^2+46*n+44)*x^3+(4*n^2+12*n+8)*x^4)*P(n,x) = -(96*n+32+(108*n^2+456*n+148)*x+(27*n^3+330*n^2+803*n+272)*x^2+(54*n^3+379*n^2+679*n+242)*x^3+(36*n^3+192*n^2+278*n+102)*x^4+(8*n^3+36*n^2+44*n+16)*x^5)*P(n-1,x) + (64+(96*n+368)*x+(36*n^2+412*n+840)*x^2+(102*n^2+682*n+984)*x^3+(106*n^2+546*n+628)*x^4+(48*n^2+212*n+208)*x^5+(8*n^2+32*n+28)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + 6*x + 3*x^2.","mathematica":"P[0, x_] := 2 + 6*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(96*n + 32 + (108*n^2 + 456*n + 148)*x + (27*n^3 + 330*n^2 + 803*n + 272)*x^2 + (54*n^3 + 379*n^2 + 679*n + 242)*x^3 + (36*n^3 + 192*n^2 + 278*n + 102)*x^4 + (8*n^3 + 36*n^2 + 44*n + 16)*x^5)*P[n - 1, x] + (64 + (96*n + 368)*x + (36*n^2 + 412*n + 840)*x^2 + (102*n^2 + 682*n + 984)*x^3 + (106*n^2 + 546*n + 628)*x^4 + (48*n^2 + 212*n + 208)*x^5 + (8*n^2 + 32*n + 28)*x^6)*D[P[n - 1, x], x])/(32 + (24*n + 92)*x + (9*n^2 + 59*n + 96)*x^2 + (12*n^2 + 46*n + 44)*x^3 + (4*n^2 + 12*n + 8)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + 6*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(96*n + 32 + (108*n^2 + 456*n + 148)*x + (27*n^3 + 330*n^2 + 803*n + 272)*x^2 + (54*n^3 + 379*n^2 + 679*n + 242)*x^3 + (36*n^3 + 192*n^2 + 278*n + 102)*x^4 + (8*n^3 + 36*n^2 + 44*n + 16)*x^5)*v[i - 1] + (64 + (96*n + 368)*x + (36*n^2 + 412*n + 840)*x^2 + (102*n^2 + 682*n + 984)*x^3 + (106*n^2 + 546*n + 628)*x^4 + (48*n^2 + 212*n + 208)*x^5 + (8*n^2 + 32*n + 28)*x^6)*deriv(v[i - 1], x))/(32 + (24*n + 92)*x + (9*n^2 + 59*n + 96)*x^2 + (12*n^2 + 46*n + 44)*x^3 + (4*n^2 + 12*n + 8)*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-5a70a6e97131d437","steps":["reverse","\u03b8+1"],"edges":["e-75e40465d2024023","e-57189fc7d6d18257"],"text":"A064861 \u2192 reverse \u2192 \u03b8+1"},"headline":"A064861 \u2192 reverse \u2192 \u03b8+1","chain":null}