{"id":"n-147f63c45421b877","revision":0,"first_index":0,"fingerprint":"f6-181e40cb14f29d52","name":"reversal 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"],["2","4"],["4","24","32"],["8","96","352","384"],["16","320","2240","6400","6144"],["32","960","10880","57600","140288","122880"],["64","2688","44800","376320","1662976","3612672","2949120"],["128","7168","164864","2007040","13862912","53788672","107053056","82575360"],["256","18432","559104","9289728","91951104","551190528","1935343616","3590848512","2642411520"],["512","46080","1781760","38707200","518332416","4412620800","23713546240","76854067200","134555369472","95126814720"],["1024","112640","5406720","148684800","2584952832","29558538240","223931924480","1102249984000","3343273426944","5572468408320","3805072588800"],["2048","270336","15769600","535265280","11712036864","172855001088","1748439531520","12057504645120","55185546149888","158248967602176","252798755143680","167423193907200"],["4096","638976","44515328","1827471360","49116807168","907887771648","11793919115264","108040106803200","689131315265536","2965404150398976","8101547955191808","12469186634711040","8036313307545600"],["8192","1490944","122257408","5969739776","193262125056","4369849516032","70798002356224","829364007600128","6996348184297472","41765636694605824","170404111731130368","446218866939396096","664470331620065280","417888291992371200"],["16384","3440640","328007680","18785894400","720829677568","19562378035200","386307577610240","5623416147148800","60437080794202112","475326771709870080","2679683878360186880","10435067990822092800","26317197211846311936","38046115154708398080","23401744351572787200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","60","840","15120","332640","8648640","259459200","8821612800","335221286400","14079294028800","647647525324800","32382376266240000","1748648318376960000","101421602465863680000"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189863,"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-d1004ba4b3ff5da5"]},{"id":189864,"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-d1004ba4b3ff5da5"]},{"id":189865,"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-d1004ba4b3ff5da5"]}],"region_exact":{},"incoming_edges":[{"id":"e-d1004ba4b3ff5da5","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-147f63c45421b877","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"d8f934b190ec8b278049d8788e990d5c4fd61cd32c13a941e9502c2807a67741","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-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-1ac4950f2738c2da","kind":"fingerprint_related","node_a_id":"n-147f63c45421b877","node_b_id":"n-ed1a169c7e7a6346","fingerprint":"f6-181e40cb14f29d52","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-181e40cb14f29d52"},"other_id":"n-ed1a169c7e7a6346","other":{"id":"n-ed1a169c7e7a6346","name":"reversal 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":"f6-181e40cb14f29d52","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["3","6"]],"card_rows":[["1"],["1"],["3","6"],["15","90","120"]],"row_sums_preview":["1","1","9","225","11025","893025","108056025","18261468225"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1336d058dca9979e","steps":["reverse"],"edges":["e-a1e397ec6cb1ca66"],"text":"A204420 \u2192 reverse"},"headline":"A204420 \u2192 reverse","chain":null}},{"id":"l-268a88b2e27a8a41","kind":"fingerprint_related","node_a_id":"n-361a7ea7d8986215","node_b_id":"n-147f63c45421b877","fingerprint":"f6-181e40cb14f29d52","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-181e40cb14f29d52"},"other_id":"n-361a7ea7d8986215","other":{"id":"n-361a7ea7d8986215","name":"mobius_x_over_x_plus_one of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","first_index":0,"fingerprint":"f6-181e40cb14f29d52","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["1","6","8"]],"card_rows":[["1"],["1","2"],["1","6","8"],["1","12","44","48"]],"row_sums_preview":["1","3","15","105","945","10395","135135","2027025"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2b477f2aa77e3efc","steps":["x \u21a6 x/(x+1)"],"edges":["e-06d460c6f7700ccf"],"text":"A109692 \u2192 x \u21a6 x/(x+1)"},"headline":"A109692 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-70b645e19c7cd7b4","kind":"fingerprint_related","node_a_id":"n-1336d058dca9979e","node_b_id":"n-147f63c45421b877","fingerprint":"f6-181e40cb14f29d52","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-181e40cb14f29d52"},"other_id":"n-1336d058dca9979e","other":{"id":"n-1336d058dca9979e","name":"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":"f6-181e40cb14f29d52","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","1"],["0","6","3"]],"card_rows":[["1"],["0","1"],["0","6","3"],["0","120","90","15"]],"row_sums_preview":["1","1","9","225","11025","893025","108056025","18261468225"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A204420"],"derivation":{"root":"n-1336d058dca9979e","steps":[],"edges":[],"text":"A204420"},"headline":"A204420: Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.","chain":null}},{"id":"l-9bd54f1792a1d8ec","kind":"fingerprint_related","node_a_id":"n-bf0f86dc9ba61887","node_b_id":"n-147f63c45421b877","fingerprint":"f6-181e40cb14f29d52","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-181e40cb14f29d52"},"other_id":"n-bf0f86dc9ba61887","other":{"id":"n-bf0f86dc9ba61887","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.","first_index":0,"fingerprint":"f6-181e40cb14f29d52","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["32","24","4"]],"card_rows":[["1"],["4","2"],["32","24","4"],["384","352","96","8"]],"row_sums_preview":["1","6","60","840","15120","332640","8648640","259459200"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9c5c208df8605708","steps":["basis A013609"],"edges":["e-8d6d917622b49386"],"text":"A225471 \u2192 basis A013609"},"headline":"A225471 \u2192 basis A013609","chain":null}},{"id":"l-cc5f4badf9ea94e0","kind":"fingerprint_related","node_a_id":"n-fdb0bda32eba4618","node_b_id":"n-147f63c45421b877","fingerprint":"f6-181e40cb14f29d52","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-181e40cb14f29d52"},"other_id":"n-fdb0bda32eba4618","other":{"id":"n-fdb0bda32eba4618","name":"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":"f6-181e40cb14f29d52","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["8","6","1"]],"card_rows":[["1"],["2","1"],["8","6","1"],["48","44","12","1"]],"row_sums_preview":["1","3","15","105","945","10395","135135","2027025"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["x \u21a6 x+1"],"edges":["e-29cb0c45f74a7f15"],"text":"A028338 \u2192 x \u21a6 x+1"},"headline":"A028338 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-e5c39e6ae29429a3","kind":"fingerprint_related","node_a_id":"n-147f63c45421b877","node_b_id":"n-6251a83089056710","fingerprint":"f6-181e40cb14f29d52","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-181e40cb14f29d52"},"other_id":"n-6251a83089056710","other":{"id":"n-6251a83089056710","name":"shift_x_plus_one of basis_map_A007318 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":"f6-181e40cb14f29d52","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["24","18","3"]],"card_rows":[["1"],["2","1"],["24","18","3"],["720","660","180","15"]],"row_sums_preview":["1","3","45","1575","99225","9823275","1404728325","273922023375"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1336d058dca9979e","steps":["basis A007318","x \u21a6 x+1"],"edges":["e-857dad5fd144a15f","e-6337179621f8c579"],"text":"A204420 \u2192 basis A007318 \u2192 x \u21a6 x+1"},"headline":"A204420 \u2192 basis A007318 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-e92193e25ffdcc9c","kind":"fingerprint_related","node_a_id":"n-f00f27ac122f92f0","node_b_id":"n-147f63c45421b877","fingerprint":"f6-181e40cb14f29d52","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-181e40cb14f29d52"},"other_id":"n-f00f27ac122f92f0","other":{"id":"n-f00f27ac122f92f0","name":"reversal 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":"f6-181e40cb14f29d52","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["1","6","8"]],"card_rows":[["1"],["1","2"],["1","6","8"],["1","12","44","48"]],"row_sums_preview":["1","3","15","105","945","10395","135135","2027025"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["x \u21a6 x+1","reverse"],"edges":["e-29cb0c45f74a7f15","e-ecaf8665ea8bb061"],"text":"A028338 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A028338 \u2192 x \u21a6 x+1 \u2192 reverse","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:   2,     4;\n  2:   4,    24,      32;\n  3:   8,    96,     352,      384;\n  4:  16,   320,    2240,     6400,      6144;\n  5:  32,   960,   10880,    57600,    140288,     122880;\n  6:  64,  2688,   44800,   376320,   1662976,    3612672,     2949120;\n  7: 128,  7168,  164864,  2007040,  13862912,   53788672,   107053056,    82575360;\n  8: 256, 18432,  559104,  9289728,  91951104,  551190528,  1935343616,  3590848512,   2642411520;\n  9: 512, 46080, 1781760, 38707200, 518332416, 4412620800, 23713546240, 76854067200, 134555369472, 95126814720;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+4*n*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[(2 + 4*n*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] = (2 + 4*n*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","reverse"],"edges":["e-8d6d917622b49386","e-d1004ba4b3ff5da5"],"text":"A225471 \u2192 basis A013609 \u2192 reverse"},"headline":"A225471 \u2192 basis A013609 \u2192 reverse","chain":null}