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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":"e33dee775dfa765a115f9d5ae317f6baaac2691ea0a804be2d4cc3050ce4fc78","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-46c95476a8505e0f","role":"sequence","name":"shift_x_plus_one of peak","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":[{"id":"l-3f41092302b9d699","kind":"fingerprint_related","node_a_id":"n-9c1df38ba8682fc5","node_b_id":"n-0f03351624a7d9e5","fingerprint":"f6-4207ccd284934a16","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4207ccd284934a16"},"other_id":"n-9c1df38ba8682fc5","other":{"id":"n-9c1df38ba8682fc5","name":"theta_plus_one_squared of shift_x_plus_one of Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= ceiling(n/2)-1) = number of permutations of [n] with k peaks.","first_index":0,"fingerprint":"f6-4207ccd284934a16","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["6","8"]],"card_rows":[["1"],["2"],["6","8"],["24","64"]],"row_sums_preview":["1","2","14","88","744","7008","76256","925312"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff270bcb802a85af","steps":["\u03b3 (scale 1/4) (shift +1)","x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-00ceaf3777abe095","e-8621fad14db1f00c","e-18cd6fdb3f8a4811"],"text":"A008292 \u2192 \u03b3 (scale 1/4) (shift +1) \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A008292 \u2192 \u03b3 (scale 1/4) (shift +1) \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}}],"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:       2;\n   3:       6,        8;\n   4:      24,       64;\n   5:     120,      480,      144;\n   6:     720,     3840,     2448;\n   7:    5040,    33600,    33264,     4352;\n   8:   40320,   322560,   435456,   126976;\n   9:  362880,  3386880,  5806080,  2703360,  198400;\n  10: 3628800, 38707200, 80559360, 52101120, 8844800;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^4-n^3-n^2+n+(4*n^4-20*n^3+36*n^2-28*n-24)*x-(9*n^4-49*n^3+83*n^2-55*n+12)*x^2)*P(n-2,x) + ((4*n^3-10*n^2+8*n+6)*x+(4*n^4-12*n^3-4*n^2+28*n+24)*x^2+(4*n^4-48*n^3+102*n^2-76*n+18)*x^3)*d/dx P(n-2,x) - ((4*n^2-8*n-4)*x^2+(8*n^3-16*n^2+8*n-8)*x^3+(8*n^3-20*n^2+16*n-4)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3-2*n^2+(4*n^3-16*n^2+20*n-8)*x)*P(n-1,x) - ((4*n^2-8*n)*x+(4*n^3-8*n^2+4*n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - n^3 - n^2 + n + (4*n^4 - 20*n^3 + 36*n^2 - 28*n - 24)*x - (9*n^4 - 49*n^3 + 83*n^2 - 55*n + 12)*x^2)*P[n - 2, x] + ((4*n^3 - 10*n^2 + 8*n + 6)*x + (4*n^4 - 12*n^3 - 4*n^2 + 28*n + 24)*x^2 + (4*n^4 - 48*n^3 + 102*n^2 - 76*n + 18)*x^3)*D[P[n - 2, x], x] - ((4*n^2 - 8*n - 4)*x^2 + (8*n^3 - 16*n^2 + 8*n - 8)*x^3 + (8*n^3 - 20*n^2 + 16*n - 4)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 - 2*n^2 + (4*n^3 - 16*n^2 + 20*n - 8)*x)*P[n - 1, x] - ((4*n^2 - 8*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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