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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":"bb42bf932a457d15ffcb28bca9113a897a6df0e4851e948606ad40c12d7ee391","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-79ebafb266da3b41","role":"sequence","name":"Even row Pascal-square read by antidiagonals.","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: 1,  8;\n  2: 1, 16,   27;\n  3: 1, 24,   90,   64;\n  4: 1, 32,  189,  320,   125;\n  5: 1, 40,  324,  896,   875,   216;\n  6: 1, 48,  495, 1920,  3150,  2016,   343;\n  7: 1, 56,  702, 3520,  8250,  9072,  4116,   512;\n  8: 1, 64,  945, 5824, 17875, 28512, 22638,  7680,   729;\n  9: 1, 72, 1224, 8960, 34125, 72072, 84084, 50688, 13365, 1000;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3+8*n^2+2*n+(48*n^4+24*n^3-8*n^2-4*n)*x+(32*n^5+32*n^4+8*n^3)*x^2)*P(n,x) = (8*n^3+8*n^2+2*n+(48*n^4+120*n^3+56*n^2-12*n-8)*x+(32*n^5+272*n^4+344*n^3-4*n^2-88*n-4)*x^2+(96*n^5+320*n^4+248*n^3-40*n^2-40*n-8)*x^3)*P(n-1,x) - ((4*n^2+4*n+1)*x+(24*n^3+64*n^2+40*n+6)*x^2+(16*n^4+160*n^3+268*n^2+124*n+12)*x^3+(64*n^4+256*n^3+304*n^2+112*n+16)*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[((8*n^3 + 8*n^2 + 2*n + (48*n^4 + 120*n^3 + 56*n^2 - 12*n - 8)*x + (32*n^5 + 272*n^4 + 344*n^3 - 4*n^2 - 88*n - 4)*x^2 + (96*n^5 + 320*n^4 + 248*n^3 - 40*n^2 - 40*n - 8)*x^3)*P[n - 1, x] - ((4*n^2 + 4*n + 1)*x + (24*n^3 + 64*n^2 + 40*n + 6)*x^2 + (16*n^4 + 160*n^3 + 268*n^2 + 124*n + 12)*x^3 + (64*n^4 + 256*n^3 + 304*n^2 + 112*n + 16)*x^4)*D[P[n - 1, x], x])/(8*n^3 + 8*n^2 + 2*n + (48*n^4 + 24*n^3 - 8*n^2 - 4*n)*x + (32*n^5 + 32*n^4 + 8*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] = ((8*n^3 + 8*n^2 + 2*n + (48*n^4 + 120*n^3 + 56*n^2 - 12*n - 8)*x + (32*n^5 + 272*n^4 + 344*n^3 - 4*n^2 - 88*n - 4)*x^2 + (96*n^5 + 320*n^4 + 248*n^3 - 40*n^2 - 40*n - 8)*x^3)*v[i - 1] - ((4*n^2 + 4*n + 1)*x + (24*n^3 + 64*n^2 + 40*n + 6)*x^2 + (16*n^4 + 160*n^3 + 268*n^2 + 124*n + 12)*x^3 + (64*n^4 + 256*n^3 + 304*n^2 + 112*n + 16)*x^4)*deriv(v[i - 1], x))/(8*n^3 + 8*n^2 + 2*n + (48*n^4 + 24*n^3 - 8*n^2 - 4*n)*x + (32*n^5 + 32*n^4 + 8*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-79ebafb266da3b41","steps":["(\u03b8+1)\u00b2"],"edges":["e-f7d0ac20788e6eb3"],"text":"A172431 \u2192 (\u03b8+1)\u00b2"},"headline":"A172431 \u2192 (\u03b8+1)\u00b2","chain":null}