{"id":"n-b8a0748f602aaf4f","revision":0,"first_index":0,"fingerprint":"f6-959faa625dfaa291","name":"theta_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":false,"rows":[["1"],["4","4"],["32","48","12"],["384","704","288","32"],["6144","12800","6720","1280","80"],["122880","280576","172800","43520","4800","192"],["2949120","7225344","4988928","1505280","224000","16128","448"],["82575360","214106112","161366016","55451648","10035200","989184","50176","1024"],["2642411520","7181697024","5806030848","2204762112","459755520","55738368","3913728","147456","2304"],["95126814720","269110738944","230562201600","94854184960","22063104000","3109994496","270950400","14254080","414720","5120"],["3805072588800","11144936816640","10029820280832","4408999936000","1119659622400","177351229440","18094669824","1189478400","48660480","1126400","11264"],["167423193907200","505597510287360","474746902806528","220742184599552","60287523225600","10490637189120","1209985007616","93696294912","4817387520","157696000","2973696","24576"],["8036313307545600","24938373269422080","24304643865575424","11861616601595904","3445656576327680","648240640819200","82557433806848","7263102173184","442051264512","18274713600","489668608","7667712","53248"],["417888291992371200","1328940663240130560","1338656600818188288","681616446924521472","208828183473029120","41978089105784832","5805548053200896","566384018849792","39328645644288","1932621250560","65667137536","1467088896","19382272","114688"],["23401744351572787200","76092230309416796160","78951591635538935808","41740271963288371200","13398419391800934400","2851960630259220480","423059565559414784","44987329177190400","3476768198492160","195623780352000","7929126453248","225430732800","4264099840","48168960","245760"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","92","1408","27024","624768","16909248","524574720","18354458880","715112663040","30705174420480","1440596611399680","73325125620633600","4024321536046694400","236907945727462195200"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-16","16","-4"],["32","-16","0"],["-16","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","4"],["8","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189850,"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-46750f40ecde5238"]},{"id":189851,"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-46750f40ecde5238"]},{"id":189852,"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-46750f40ecde5238"]},{"id":189853,"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-46750f40ecde5238"]}],"region_exact":{},"incoming_edges":[{"id":"e-46750f40ecde5238","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-b8a0748f602aaf4f","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. 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":"1431739ac0eb96087f3c08a13133bf571403477c88d650c21e88caa54d4f19d1","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]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-ad55731336b07c52","kind":"fingerprint_related","node_a_id":"n-b8a0748f602aaf4f","node_b_id":"n-df9ff482d15e6222","fingerprint":"f6-959faa625dfaa291","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-959faa625dfaa291"},"other_id":"n-df9ff482d15e6222","other":{"id":"n-df9ff482d15e6222","name":"derivative of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.","first_index":0,"fingerprint":"f6-959faa625dfaa291","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["6","6"]],"card_rows":[["1"],["6","6"],["120","180","45"],["5040","9240","3780","420"]],"row_sums_preview":["0","1","12","345","18480","1596105","202951980","35703612945"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1336d058dca9979e","steps":["D"],"edges":["e-5f410c7b8b40e256"],"text":"A204420 \u2192 D"},"headline":"A204420 \u2192 D","chain":null}},{"id":"l-d2b377a7ee71129d","kind":"fingerprint_related","node_a_id":"n-8802fa67ca7b6a9f","node_b_id":"n-b8a0748f602aaf4f","fingerprint":"f6-959faa625dfaa291","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-959faa625dfaa291"},"other_id":"n-8802fa67ca7b6a9f","other":{"id":"n-8802fa67ca7b6a9f","name":"theta_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":"f6-959faa625dfaa291","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["8","12","3"]],"card_rows":[["1"],["2","2"],["8","12","3"],["48","88","36","4"]],"row_sums_preview":["1","4","23","176","1689","19524","264207","4098240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-29cb0c45f74a7f15","e-6b6fe28ce0cbc59d"],"text":"A028338 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A028338 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","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:           4,            4;\n  2:          32,           48,           12;\n  3:         384,          704,          288,          32;\n  4:        6144,        12800,         6720,        1280,          80;\n  5:      122880,       280576,       172800,       43520,        4800,        192;\n  6:     2949120,      7225344,      4988928,     1505280,      224000,      16128,       448;\n  7:    82575360,    214106112,    161366016,    55451648,    10035200,     989184,     50176,     1024;\n  8:  2642411520,   7181697024,   5806030848,  2204762112,   459755520,   55738368,   3913728,   147456,   2304;\n  9: 95126814720, 269110738944, 230562201600, 94854184960, 22063104000, 3109994496, 270950400, 14254080, 414720, 5120;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-32*n+16+(16*n-16)*x+4*x^2)*P(n-2,x) + (8*n-4+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 32*n + 16 + (16*n - 16)*x + 4*x^2)*P[n - 2, x] + (8*n - 4 + 4*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 32*n + 16 + (16*n - 16)*x + 4*x^2)*v[i - 2] + (8*n - 4 + 4*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","\u03b8+1"],"edges":["e-8d6d917622b49386","e-46750f40ecde5238"],"text":"A225471 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A225471 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}
