{"id":"n-a33b47c8b957d83a","revision":1,"first_index":0,"fingerprint":"f6-9e437d53009ce099","name":"basis_map_A013609 of Triangle read by rows, s_3(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"],["3","2"],["18","18","4"],["162","198","72","8"],["1944","2700","1260","240","16"],["29160","44388","24300","6120","720","32"],["524880","857304","526176","158760","25200","2016","64"],["11022480","19053144","12764304","4386312","846720","92736","5376","128"],["264539520","479320416","344449584","130800096","29093904","3919104","314496","13824","256"],["7142567040","13470730272","10258779600","4220501760","1047135600","164003616","16329600","1002240","34560","512"],["214277011200","418407042240","334704848544","147132612000","39855071520","7014379680","817895232","62726400","3041280","84480","1024"],["7071141369600","14235986416320","11882074086432","5524785893088","1609482584160","311184672480","41019302016","3705761664","225815040","8870400","202752","2048"],["254561089305600","526637793726720","456226639944192","222656440324032","68990944815936","14421613377600","2099064217536","215446023936","15540864768","770964480","25039872","479232","4096"],["9927882482918400","21047996133953280","18846114545276928","9596054452525632","3135959728469568","700424811358272","110706731239104","12600523368576","1036985773824","61149344256","2518483968","68769792","1118208","8192"],["416971064282572800","903871602591874560","833632803169537536","440726516096630400","150902417500773120","35689761533986560","6050532334758912","750635443958400","68754449237760","4642244006400","228075015168","7925299200","184504320","2580480","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","40","440","6160","104720","2094400","48171200","1252451200","36321084800","1162274713600","40679614976000","1545825369088000","63378840132608000","2788668965834752000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166795,"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-0826250557dc6011"]},{"id":166796,"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-0826250557dc6011"]},{"id":166797,"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-0826250557dc6011"]}],"region_exact":{},"incoming_edges":[{"id":"e-0826250557dc6011","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-a33b47c8b957d83a","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":"8723e6381b627a4bd4c0b301812bdf5296dd3758dee76b798c6ffbf2f6f223da","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-8229dbdd6a4b5cd9","role":"sequence","name":"Triangle read by rows, s_3(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-63024cf163ca4ab5","kind":"oeis_match","node_a_id":"n-a33b47c8b957d83a","node_b_id":null,"fingerprint":null,"oeis_id":"A034000","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A034000","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A034000","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-a33b47c8b957d83a","oeis_id":"A034000","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":19,"oeis_terms_available":19,"term_count":200,"raw_size":43820}],"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:          3,           2;\n  2:         18,          18,           4;\n  3:        162,         198,          72,          8;\n  4:       1944,        2700,        1260,        240,         16;\n  5:      29160,       44388,       24300,       6120,        720,        32;\n  6:     524880,      857304,      526176,     158760,      25200,      2016,       64;\n  7:   11022480,    19053144,    12764304,    4386312,     846720,     92736,     5376,     128;\n  8:  264539520,   479320416,   344449584,  130800096,   29093904,   3919104,   314496,   13824,   256;\n  9: 7142567040, 13470730272, 10258779600, 4220501760, 1047135600, 164003616, 16329600, 1002240, 34560, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n+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[(3*n + 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] = (3*n + 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-8229dbdd6a4b5cd9","steps":["basis A013609"],"edges":["e-0826250557dc6011"],"text":"A225470 \u2192 basis A013609"},"headline":"A225470 \u2192 basis A013609","chain":null}
