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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":"04d324d01d9d90e36960abb01651108a60b2168ed0b788695b9e0790adcd4edb","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-0e6d847331141ca6","role":"sequence","name":"basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1","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,    8;\n  2:  3,   32,    27;\n  3:  4,   80,   180,     64;\n  4:  5,  160,   675,    640,    125;\n  5:  6,  280,  1890,   3360,   1750,    216;\n  6:  7,  448,  4410,  12544,  12250,   4032,    343;\n  7:  8,  672,  9072,  37632,  58800,  36288,   8232,    512;\n  8:  9,  960, 17010,  96768, 220500, 217728,  92610,  15360,   729;\n  9: 10, 1320, 29700, 221760, 693000, 997920, 679140, 211200, 26730, 1000;\n","recurrence_sentence":"The row polynomials satisfy (n^2+4*n+(2*n^4+10*n^3+6*n^2-8*n)*x+(n^6+8*n^5+20*n^4+16*n^3)*x^2)*P(n,x) = (n^2+5*n+4+(2*n^4+12*n^3+28*n^2+18*n)*x+(n^6+9*n^5+37*n^4+69*n^3+47*n^2+7*n)*x^2+(3*n^6+25*n^5+77*n^4+113*n^3+92*n^2+42*n+8)*x^3)*P(n-1,x) + ((2*n+2)*x+(4*n^3+14*n^2+10*n)*x^2+(2*n^5+16*n^4+42*n^3+44*n^2+22*n+6)*x^3-(2*n^5+16*n^4+46*n^3+58*n^2+34*n+8)*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[((n^2 + 5*n + 4 + (2*n^4 + 12*n^3 + 28*n^2 + 18*n)*x + (n^6 + 9*n^5 + 37*n^4 + 69*n^3 + 47*n^2 + 7*n)*x^2 + (3*n^6 + 25*n^5 + 77*n^4 + 113*n^3 + 92*n^2 + 42*n + 8)*x^3)*P[n - 1, x] + ((2*n + 2)*x + (4*n^3 + 14*n^2 + 10*n)*x^2 + (2*n^5 + 16*n^4 + 42*n^3 + 44*n^2 + 22*n + 6)*x^3 - (2*n^5 + 16*n^4 + 46*n^3 + 58*n^2 + 34*n + 8)*x^4)*D[P[n - 1, x], x])/(n^2 + 4*n + (2*n^4 + 10*n^3 + 6*n^2 - 8*n)*x + (n^6 + 8*n^5 + 20*n^4 + 16*n^3)*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] = ((n^2 + 5*n + 4 + (2*n^4 + 12*n^3 + 28*n^2 + 18*n)*x + (n^6 + 9*n^5 + 37*n^4 + 69*n^3 + 47*n^2 + 7*n)*x^2 + (3*n^6 + 25*n^5 + 77*n^4 + 113*n^3 + 92*n^2 + 42*n + 8)*x^3)*v[i - 1] + ((2*n + 2)*x + (4*n^3 + 14*n^2 + 10*n)*x^2 + (2*n^5 + 16*n^4 + 42*n^3 + 44*n^2 + 22*n + 6)*x^3 - (2*n^5 + 16*n^4 + 46*n^3 + 58*n^2 + 34*n + 8)*x^4)*deriv(v[i - 1], x))/(n^2 + 4*n + (2*n^4 + 10*n^3 + 6*n^2 - 8*n)*x + (n^6 + 8*n^5 + 20*n^4 + 16*n^3)*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-0e6d847331141ca6","steps":["(\u03b8+1)\u00b2"],"edges":["e-915e0a73a4c39c54"],"text":"A145596 \u2192 (\u03b8+1)\u00b2"},"headline":"A145596 \u2192 (\u03b8+1)\u00b2","chain":null}