{"id":"n-843f07b55ed17e8a","revision":1,"first_index":0,"fingerprint":"f6-f68e3e3b9cc84500","name":"basis_map_A007318 of Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","1"],["24","11","1"],["192","122","21","1"],["1920","1514","371","34","1"],["23040","21264","6610","875","50","1"],["322560","336216","124594","21195","1765","69","1"],["5160960","5933136","2522044","517384","55825","3199","91","1"],["92897280","115833168","55029708","13082804","1715329","128072","5362","116","1"],["1857945600","2481854400","1293490656","346844924","53073216","4840689","265104","8466","144","1"],["40874803200","57945461760","32674717296","9690778580","1683355400","180081615","12093753","506670","12750","175","1"],["980995276800","1465012535040","884317905504","285855305736","55248754660","6741178730","532946337","27458013","908490","18480","209","1"],["25505877196800","39888857168640","25558649672544","8902598672616","1885653402556","257130821310","23267442407","1415131938","57706143","1546050","25949","246","1"],["714164561510400","1163976253009920","786327657553152","292475741187552","67091123320936","10067638491196","1021549260446","71305819871","3439928778","113782383","2518802","35477","286","1"],["21424936845312000","36245599431552000","25672717046989440","10122238611785664","2491016205457416","406403959485524","45524907654226","3572591489467","198217721273","7769809047","212645433","3954769","47411","329","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","36","336","3840","51840","806400","14192640","278691840","6038323200","143061811200","3678732288000","102023508787200","3035199386419200","96412215803904000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0"],["2"],["-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","1"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161953,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 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-021513687e7cc411"]},{"id":161954,"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-021513687e7cc411"]},{"id":161955,"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-021513687e7cc411"]},{"id":161956,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-021513687e7cc411"]}],"region_exact":{},"incoming_edges":[{"id":"e-021513687e7cc411","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-843f07b55ed17e8a","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"449c95f9807fb9575d5935043b7019894abb83e818bbb991da6d18a7ece74961","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6ace63f6e12c5046","role":"sequence","name":"Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.","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:          4,          1;\n  2:         24,         11,          1;\n  3:        192,        122,         21,         1;\n  4:       1920,       1514,        371,        34,        1;\n  5:      23040,      21264,       6610,       875,       50,       1;\n  6:     322560,     336216,     124594,     21195,     1765,      69,      1;\n  7:    5160960,    5933136,    2522044,    517384,    55825,    3199,     91,    1;\n  8:   92897280,  115833168,   55029708,  13082804,  1715329,  128072,   5362,  116,   1;\n  9: 1857945600, 2481854400, 1293490656, 346844924, 53073216, 4840689, 265104, 8466, 144, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2*n^2-2*n)*P(n-2,x) + (3*n+1+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[-(2*n^2 - 2*n)*P[n - 2, x] + (3*n + 1 + x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = -(2*n^2 - 2*n)*v[i - 2] + (3*n + 1 + x)*v[i - 1]); 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-6ace63f6e12c5046","steps":["basis A007318"],"edges":["e-021513687e7cc411"],"text":"A079638 \u2192 basis A007318"},"headline":"A079638 \u2192 basis A007318","chain":null}