{"id":"n-d3e87c9d56ff1da1","revision":0,"first_index":0,"fingerprint":"f6-e2f382b41485ddb6","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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":true,"rows":[["1"],["6","2"],["60","32","4"],["840","568","120","8"],["15120","11904","3296","384","16"],["332640","292128","96320","15040","1120","32"],["8648640","8260608","3088576","583680","59200","3072","64"],["259459200","265115520","109178496","23687552","2943360","210560","8064","128"],["8821612800","9532846080","4242299904","1023733760","147449344","13045760","695296","20480","256"],["335221286400","379891376640","180273088512","47386482688","7650542592","790637568","52512768","2168832","50688","512"],["14079294028800","16625880391680","8331252470784","2350778449920","416095754240","48507863040","3786811392","196116480","6466560","122880","1024"],["647647525324800","792949086074880","416489374439424","124798313637888","23841961594880","3063553208320","271209050112","16594980864","689694720","18585600","292864","2048"],["32382376266240000","40942749354393600","22410366894120960","7072894430773248","1441694707019776","200861583605760","19687558922240","1372167143424","67674697728","2308669440","51814400","688128","4096"],["1748648318376960000","2275673217669734400","1292045310991319040","426757033049997312","91997303040614400","13729914928750592","1464851349012480","113472143589376","6398767964160","260017545216","7415316480","140787712","1597440","8192"],["101421602465863680000","135486343261598515200","79489974472835973120","27335998538882482176","6189357642455629824","980329671948763136","112421208100225024","9511087026208768","598072829100032","27878553550848","950123446272","22996320256","374226944","3670016","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","96","1536","30720","737280","20643840","660602880","23781703680","951268147200","41855798476800","2009078326886400","104472072998092800","5850436087893196800","351026165273591808000"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189858,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-72a49fb9f450a2b3"]},{"id":189859,"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-4f6f9aa969b5ed9d","e-72a49fb9f450a2b3"]},{"id":189860,"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-4f6f9aa969b5ed9d","e-72a49fb9f450a2b3"]},{"id":189861,"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-4f6f9aa969b5ed9d","e-72a49fb9f450a2b3"]},{"id":189862,"position":4,"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-4f6f9aa969b5ed9d"]}],"region_exact":{},"incoming_edges":[{"id":"e-4f6f9aa969b5ed9d","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-d3e87c9d56ff1da1","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":"fda2ca39bb527571340532a5183aca4ac3608dd45bfd8395021dea2a26b2e3e7","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-bf0f86dc9ba61887","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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]"}]},{"id":"e-72a49fb9f450a2b3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-d3e87c9d56ff1da1","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"0e642623d5ba6ed52855fa396d9862a284297d62b548a0fd883d650ca5ba3455","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-bf0f86dc9ba61887","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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-a248b9e2b1676837","kind":"fingerprint_related","node_a_id":"n-d371108fa3eab9d3","node_b_id":"n-d3e87c9d56ff1da1","fingerprint":"f6-e2f382b41485ddb6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e2f382b41485ddb6"},"other_id":"n-d371108fa3eab9d3","other":{"id":"n-d371108fa3eab9d3","name":"shift_x_plus_one of basis_map_A007318 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-e2f382b41485ddb6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","1"],["15","8","1"]],"card_rows":[["1"],["3","1"],["15","8","1"],["105","71","15","1"]],"row_sums_preview":["1","4","24","192","1920","23040","322560","5160960"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["x \u21a6 x+1","basis A007318"],"edges":["e-29cb0c45f74a7f15","e-acfa906a567cd413"],"text":"A028338 \u2192 x \u21a6 x+1 \u2192 basis A007318"},"headline":"A028338 \u2192 x \u21a6 x+1 \u2192 basis A007318","chain":null}}],"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,            2;\n  2:           60,           32,            4;\n  3:          840,          568,          120,           8;\n  4:        15120,        11904,         3296,         384,         16;\n  5:       332640,       292128,        96320,       15040,       1120,        32;\n  6:      8648640,      8260608,      3088576,      583680,      59200,      3072,       64;\n  7:    259459200,    265115520,    109178496,    23687552,    2943360,    210560,     8064,     128;\n  8:   8821612800,   9532846080,   4242299904,  1023733760,  147449344,  13045760,   695296,   20480,   256;\n  9: 335221286400, 379891376640, 180273088512, 47386482688, 7650542592, 790637568, 52512768, 2168832, 50688, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+2+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(4*n + 2 + 2*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; for(n = 1, N, my(i = n + 1); v[i] = (4*n + 2 + 2*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-9c5c208df8605708","steps":["basis A013609","basis A007318"],"edges":["e-8d6d917622b49386","e-4f6f9aa969b5ed9d"],"text":"A225471 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A225471 \u2192 basis A013609 \u2192 basis A007318","chain":null}