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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-29e8dbf6785f009b"]},{"id":166197,"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-29e8dbf6785f009b"]},{"id":166198,"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 real-rootedness)","region_exact":null,"edges":["e-29e8dbf6785f009b"]}],"region_exact":{},"incoming_edges":[{"id":"e-29e8dbf6785f009b","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-e143e6e88ae14e54","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"d2429fb6b71fbdd7a3bff8281cd3a0d8cf5e0cf1179c2a164665930b81af80b9","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-344e4bfa9933088a","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.","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:             529,               627,                121;\n  1:           12167,             27137,              12342,                1331;\n  2:          279841,           1056495,             853897,              210298,               14641;\n  3:         6436343,          38999081,           50046810,            21169555,             3294225,              161051;\n  4:       148035889,        1396768527,         2681096341,          1736009990,           453578180,            48798453,            1771561;\n  5:      3404825447,       49118524697,       136013789142,        126708619191,         49485847950,          8795639314,          694451912,           19487171;\n  6:     78310985281,     1707482919615,      6660924283057,       8591834355258,       4709346623751,       1230404193942,       158558252622,         9587688132,         214358881;\n  7:   1801152661463,    58915840105001,    318523904853330,     554412891008075,     410036401699155,     147774339988737,     27646130616720,      2702909592042,      129258405243,      2357947691;\n  8:  41426511213649,  2022951242846127,  14981649959554861,   34550884849838830,   33570980714932210,   16036798937812191,   4086023364764187,    574398395988120,    44079688494435,   1709512075975,    25937424601;\n  9: 952809757913927, 69236033880118457, 696426711851276142, 2099647701146077951, 2629315441248685200, 1620151105013605208, 540060867779946744, 102386096611218057, 11211227778751605, 693437046707735, 22254310307658, 285311670611;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (23+11*x)*P(n-1,x) + 11*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 529 + 627*x + 121*x^2.","mathematica":"P[0, x_] := 529 + 627*x + 121*x^2;\nP[n_, x_] := P[n, x] = Expand[(23 + 11*x)*P[n - 1, x] + 11*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 529 + 627*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (23 + 11*x)*v[i - 1] + 11*x*deriv(v[i - 1], x)); 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-344e4bfa9933088a","steps":["basis A008277"],"edges":["e-29e8dbf6785f009b"],"text":"A038337 \u2192 basis A008277"},"headline":"A038337 \u2192 basis A008277","chain":null}