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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":"2cb3f4acd46fb3390b873a0fd7c02d85dea64df54dc8134546ad63ac7193367a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-047228a38014e6d4","role":"sequence","name":"Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","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":[{"id":"l-1c1648b977b2aa9b","kind":"oeis_match","node_a_id":"n-074c6703c2bee119","node_b_id":null,"fingerprint":null,"oeis_id":"A350265","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A350265","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A350265","relation":"match_row_sums"}],"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,   2;\n  2: 1,   8,     3;\n  3: 1,  20,    30,      4;\n  4: 1,  40,   150,     80,      5;\n  5: 1,  70,   525,    700,    175,      6;\n  6: 1, 112,  1470,   3920,   2450,    336,      7;\n  7: 1, 168,  3528,  16464,  20580,   7056,    588,     8;\n  8: 1, 240,  7560,  56448, 123480,  84672,  17640,   960,    9;\n  9: 1, 330, 14850, 166320, 582120, 698544, 291060, 39600, 1485, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^5+10*n^4+35*n^3+50*n^2+24*n)*P(n,x) = (n^5+10*n^4+35*n^3+50*n^2+24*n+(10*n^5+55*n^4+98*n^3+65*n^2+12*n)*x)*P(n-1,x) + ((3*n^4+33*n^3+108*n^2+123*n+36)*x-(15*n^4+69*n^3+87*n^2+24*n)*x^2)*d/dx P(n-1,x) + ((6*n^3+28*n^2+37*n+12)*x^2+(6*n^3+28*n^2+37*n+12)*x^3)*d^2/dx^2 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^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (10*n^5 + 55*n^4 + 98*n^3 + 65*n^2 + 12*n)*x)*P[n - 1, x] + ((3*n^4 + 33*n^3 + 108*n^2 + 123*n + 36)*x - (15*n^4 + 69*n^3 + 87*n^2 + 24*n)*x^2)*D[P[n - 1, x], x] + ((6*n^3 + 28*n^2 + 37*n + 12)*x^2 + (6*n^3 + 28*n^2 + 37*n + 12)*x^3)*D[P[n - 1, x], {x, 2}])/(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)]];\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^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (10*n^5 + 55*n^4 + 98*n^3 + 65*n^2 + 12*n)*x)*v[i - 1] + ((3*n^4 + 33*n^3 + 108*n^2 + 123*n + 36)*x - (15*n^4 + 69*n^3 + 87*n^2 + 24*n)*x^2)*deriv(v[i - 1], x) + ((6*n^3 + 28*n^2 + 37*n + 12)*x^2 + (6*n^3 + 28*n^2 + 37*n + 12)*x^3)*deriv(deriv(v[i - 1], x), x))/(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)); 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-c963abfc0266d66c","steps":["\u2299 C(n+1,k+1) (shift +1)","\u03b8+1"],"edges":["e-548c39a476f786f5","e-f6a16136bb4575f0"],"text":"A108838 \u2192 \u2299 C(n+1,k+1) (shift +1) \u2192 \u03b8+1"},"headline":"A108838 \u2192 \u2299 C(n+1,k+1) (shift +1) \u2192 \u03b8+1","chain":null}