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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":"9552f5ff7bcf41996b51ff0628a0882ccbc17495a587a6176ce15e19f015b215","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-c30cb06ad75c5e21","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(2n+1, 2k+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:  3,    4;\n  2:  5,   40,      9;\n  3:  7,  140,    189,     16;\n  4:  9,  336,   1134,    576,      25;\n  5: 11,  660,   4158,   5280,    1375,      36;\n  6: 13, 1144,  11583,  27456,   17875,    2808,      49;\n  7: 15, 1820,  27027, 102960,  125125,   49140,    5145,     64;\n  8: 17, 2720,  55692, 311168,  607750,  445536,  116620,   8704,    81;\n  9: 19, 3876, 104652, 806208, 2309450, 2720952, 1329468, 248064, 13851, 100;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1-(20*n^3-28*n^2+11*n-1)*x+(8*n^5-12*n^4+6*n^3-n^2)*x^2)*P(n,x) = (2*n+1-(20*n^3-8*n^2-31*n+21)*x+(8*n^5-4*n^4-110*n^3+49*n^2+89*n-37)*x^2+(24*n^5+20*n^4-38*n^3-13*n^2+14*n-3)*x^3)*P(n-1,x) + (4*x-(40*n^2+24*n-10)*x^2+(16*n^4+32*n^3+44*n^2+16*n-12)*x^3-(16*n^4+32*n^3+4*n^2-8*n+2)*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[((2*n + 1 - (20*n^3 - 8*n^2 - 31*n + 21)*x + (8*n^5 - 4*n^4 - 110*n^3 + 49*n^2 + 89*n - 37)*x^2 + (24*n^5 + 20*n^4 - 38*n^3 - 13*n^2 + 14*n - 3)*x^3)*P[n - 1, x] + (4*x - (40*n^2 + 24*n - 10)*x^2 + (16*n^4 + 32*n^3 + 44*n^2 + 16*n - 12)*x^3 - (16*n^4 + 32*n^3 + 4*n^2 - 8*n + 2)*x^4)*D[P[n - 1, x], x])/(2*n - 1 - (20*n^3 - 28*n^2 + 11*n - 1)*x + (8*n^5 - 12*n^4 + 6*n^3 - 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; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 1 - (20*n^3 - 8*n^2 - 31*n + 21)*x + (8*n^5 - 4*n^4 - 110*n^3 + 49*n^2 + 89*n - 37)*x^2 + (24*n^5 + 20*n^4 - 38*n^3 - 13*n^2 + 14*n - 3)*x^3)*v[i - 1] + (4*x - (40*n^2 + 24*n - 10)*x^2 + (16*n^4 + 32*n^3 + 44*n^2 + 16*n - 12)*x^3 - (16*n^4 + 32*n^3 + 4*n^2 - 8*n + 2)*x^4)*deriv(v[i - 1], x))/(2*n - 1 - (20*n^3 - 28*n^2 + 11*n - 1)*x + (8*n^5 - 12*n^4 + 6*n^3 - 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-c30cb06ad75c5e21","steps":["(\u03b8+1)\u00b2"],"edges":["e-a07ebdda950bedd8"],"text":"A103327 \u2192 (\u03b8+1)\u00b2"},"headline":"A103327 \u2192 (\u03b8+1)\u00b2","chain":null}