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0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-a3757d9096387327"]},{"id":192048,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-a3757d9096387327"]},{"id":192049,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve 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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":"5b7ad5835efb1c84d66b532a3b22e07ed51746a46006f0beb9ac8424193656b6","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-83675781d9687218","role":"sequence","name":"basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","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:        6,        28,         24;\n  2:       36,       336,        876,         896,          320;\n  3:      216,      3024,      13176,       27104,        29280,         16128,          3584;\n  4:     1296,     24192,     147744,      456960,       825680,        913920,        612864,        229376,          36864;\n  5:     7776,    181440,    1425600,     5886720,     14920800,      24838464,      27852160,      20930560,       10137600,        2867200,         360448;\n  6:    46656,   1306368,   12550464,    64834560,    211550400,     470340864,     739920832,     836161536,      676961280,      384204800,      145440768,       33030144,        3407872;\n  7:   279936,   9144576,  103856256,   644184576,   2571246720,    7154106624,   14491123584,   21872671744,    24841926144,    21197352960,    13408575488,     6107824128,     1896480768,     359661568,      31457280;\n  8:  1679616,  62705664,  821892096,  5950070784,  28085460480,   93851799552,  232595112960,  439404478464,   642597693696,   732340797440,   649789521920,   444927049728,   230786203648,   87757422592,   23089643520,    3758096384,    285212672;\n  9: 10077696, 423263232, 6288482304, 52045701120, 283597562880, 1106347382784, 3243040561152, 7359034540032, 13168613574144, 18784053693440, 21459962861568, 19624092106752, 14276242046976, 8158742839296, 3585233387520, 1169707499520, 266959060992, 38050725888, 2550136832;\n","recurrence_sentence":"The row polynomials satisfy (3+7*n*x+(8*n-4)*x^2)*P(n,x) = (18+(42*n+84)*x+(146*n+146)*x^2+(168*n+112)*x^3+(64*n+32)*x^4)*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[((18 + (42*n + 84)*x + (146*n + 146)*x^2 + (168*n + 112)*x^3 + (64*n + 32)*x^4)*P[n - 1, x])/(3 + 7*n*x + (8*n - 4)*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] = ((18 + (42*n + 84)*x + (146*n + 146)*x^2 + (168*n + 112)*x^3 + (64*n + 32)*x^4)*v[i - 1])/(3 + 7*n*x + (8*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-eed4f744ce063213","steps":["basis A013609","\u03b8+1"],"edges":["e-185113cbc4b6c77b","e-a3757d9096387327"],"text":"A200536 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A200536 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}