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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":"398bab9b84b85fe94607f2a0a338c06b18e4bc0beeba8dad58be402953ae4d2e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4689fb1386d440ad","role":"sequence","name":"T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,   8;\n   3: 1,  24,    54;\n   4: 1,  48,   270,    320;\n   5: 1,  80,   810,   2240,    1750;\n   6: 1, 120,  1890,   8960,   15750,    9072;\n   7: 1, 168,  3780,  26880,   78750,   99792,    45276;\n   8: 1, 224,  6804,  67200,  288750,  598752,   588588,   219648;\n   9: 1, 288, 11340, 147840,  866250, 2594592,  4120116,  3294720,  1042470;\n  10: 1, 360, 17820, 295680, 2252250, 9081072, 20600580, 26357760, 17721990, 4862000;\n","recurrence_sentence":"The row polynomials satisfy (n-1-(4*n^3-16*n^2+21*n-9)*x+(n^5-7*n^4+19*n^3-25*n^2+16*n-4)*x^2)*P(n,x) = (n-1-(4*n^3-16*n^2+13*n-1)*x+(n^5-7*n^4+5*n^3+17*n^2-20*n+4)*x^2+(2*n^5-10*n^4+16*n^3-8*n^2)*x^3)*P(n-1,x) + (2*x-(8*n^2-16*n+2)*x^2+(2*n^4-8*n^3+16*n-4)*x^3+(2*n^4-8*n^3+8*n^2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 1 - (4*n^3 - 16*n^2 + 13*n - 1)*x + (n^5 - 7*n^4 + 5*n^3 + 17*n^2 - 20*n + 4)*x^2 + (2*n^5 - 10*n^4 + 16*n^3 - 8*n^2)*x^3)*P[n - 1, x] + (2*x - (8*n^2 - 16*n + 2)*x^2 + (2*n^4 - 8*n^3 + 16*n - 4)*x^3 + (2*n^4 - 8*n^3 + 8*n^2)*x^4)*D[P[n - 1, x], x])/(n - 1 - (4*n^3 - 16*n^2 + 21*n - 9)*x + (n^5 - 7*n^4 + 19*n^3 - 25*n^2 + 16*n - 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n - 1 - (4*n^3 - 16*n^2 + 13*n - 1)*x + (n^5 - 7*n^4 + 5*n^3 + 17*n^2 - 20*n + 4)*x^2 + (2*n^5 - 10*n^4 + 16*n^3 - 8*n^2)*x^3)*v[i - 1] + (2*x - (8*n^2 - 16*n + 2)*x^2 + (2*n^4 - 8*n^3 + 16*n - 4)*x^3 + (2*n^4 - 8*n^3 + 8*n^2)*x^4)*deriv(v[i - 1], x))/(n - 1 - (4*n^3 - 16*n^2 + 21*n - 9)*x + (n^5 - 7*n^4 + 19*n^3 - 25*n^2 + 16*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4689fb1386d440ad","steps":["(\u03b8+1)\u00b2"],"edges":["e-4e0fa4d6f32bf88e"],"text":"A063007 \u2192 (\u03b8+1)\u00b2"},"headline":"A063007 \u2192 (\u03b8+1)\u00b2","chain":null}