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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. 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":"611835ab81905ef0b39ba9204642b523ebecf4c6526d2528b1369aa06487ce3c","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-7d332fcb7bf9a0f7","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).","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;\n  2:  2,   2;\n  3:  3,   4;\n  4:  5,  10,   3;\n  5:  8,  20,   9;\n  6: 13,  40,  27,   4;\n  7: 21,  76,  66,  16;\n  8: 34, 142, 153,  56,  5;\n  9: 55, 260, 333, 160, 25;\n","recurrence_sentence":"The row polynomials satisfy (10*n+(5*n^2+7*n)*x+(n^3+n^2)*x^2)*P(n,x) = (30*n-20+(20*n^2+36*n-44)*x+(3*n^3+21*n^2+6*n-28)*x^2+(2*n^3+4*n^2-2*n-4)*x^3)*P(n-1,x) - (25+(25*n+80)*x+(5*n^2+60*n+93)*x^2+(9*n^2+47*n+46)*x^3+(4*n^2+12*n+8)*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[((30*n - 20 + (20*n^2 + 36*n - 44)*x + (3*n^3 + 21*n^2 + 6*n - 28)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*P[n - 1, x] - (25 + (25*n + 80)*x + (5*n^2 + 60*n + 93)*x^2 + (9*n^2 + 47*n + 46)*x^3 + (4*n^2 + 12*n + 8)*x^4)*D[P[n - 1, x], x])/(10*n + (5*n^2 + 7*n)*x + (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] = ((30*n - 20 + (20*n^2 + 36*n - 44)*x + (3*n^3 + 21*n^2 + 6*n - 28)*x^2 + (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*v[i - 1] - (25 + (25*n + 80)*x + (5*n^2 + 60*n + 93)*x^2 + (9*n^2 + 47*n + 46)*x^3 + (4*n^2 + 12*n + 8)*x^4)*deriv(v[i - 1], x))/(10*n + (5*n^2 + 7*n)*x + (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-75ccd0453a9b6cec","steps":["basis A007318","\u03b8+1"],"edges":["e-2a29a8b9938e2bd7","e-eed06076b0f64cdc"],"text":"A011973 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A011973 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}