{"id":"n-5b94261d6f603245","revision":1,"first_index":0,"fingerprint":"f6-34eaf45657600f7a","name":"basis_map_A013610 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","3"],["18","27","9"],["162","297","162","27"],["1944","4050","2835","810","81"],["29160","66582","54675","20655","3645","243"],["524880","1285956","1183896","535815","127575","15309","729"],["11022480","28579716","28719684","14803803","4286520","704214","61236","2187"],["264539520","718980624","775011564","441450324","147287889","29760696","3582306","236196","6561"],["7142567040","20206095408","23082254100","14244193440","5301123975","1245402459","186004350","17124210","885735","19683"],["214277011200","627610563360","753085909224","496572565500","201766299570","53265445695","9316337877","1071739350","77944680","3247695","59049"],["7071141369600","21353979624480","26734666694472","18646152389172","8148005582310","2363058606645","467235487026","63316412181","5787392490","341007975","11691702","177147"],["254561089305600","789956690590080","1026509939874432","751465486093608","349266658130676","109514126586150","23909653352871","3681097299594","398295366183","29638464570","1443925197","41452398","531441"],["9927882482918400","31571994200929920","42403757726873088","32386683777274008","15875796125377188","5318850911251878","1261018860520419","215291754742779","26576811179919","2350786216779","145228476393","5948419113","145083393","1594323"],["416971064282572800","1355807403887811840","1875673807131459456","1487451991826127600","763943488597663920","271019126648710440","68919344875613232","12825310280757975","1762101333784935","178463454644475","13151954658843","685519031925","23938759845","502211745","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","54","648","9720","174960","3674160","88179840","2380855680","71425670400","2357047123200","84853696435200","3309294160972800","138990354760857600","6254565964238592000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","3"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167490,"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-86f8b58a2aa07621"]},{"id":167491,"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-86f8b58a2aa07621"]},{"id":167492,"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-86f8b58a2aa07621"]}],"region_exact":{},"incoming_edges":[{"id":"e-86f8b58a2aa07621","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-5b94261d6f603245","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":"cd4630f3434a2d513a33db032834e91fba731b8339d9eb9020706b6df81e3e9c","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-07ccadf1888ef8cd","kind":"fingerprint_related","node_a_id":"n-efe23a76953cf69d","node_b_id":"n-5b94261d6f603245","fingerprint":"f6-34eaf45657600f7a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-34eaf45657600f7a"},"other_id":"n-efe23a76953cf69d","other":{"id":"n-efe23a76953cf69d","name":"Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","first_index":0,"fingerprint":"f6-34eaf45657600f7a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["2","3","1"]],"card_rows":[["1"],["1","1"],["2","3","1"],["6","11","6","1"]],"row_sums_preview":["1","2","6","24","120","720","5040","40320"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A130534"],"derivation":{"root":"n-efe23a76953cf69d","steps":[],"edges":[],"text":"A130534"},"headline":"A130534: Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","chain":null}},{"id":"l-4e3c43c53f2cbd1d","kind":"fingerprint_related","node_a_id":"n-b9fbfb57d9269e2f","node_b_id":"n-5b94261d6f603245","fingerprint":"f6-34eaf45657600f7a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-34eaf45657600f7a"},"other_id":"n-b9fbfb57d9269e2f","other":{"id":"n-b9fbfb57d9269e2f","name":"Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-34eaf45657600f7a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","1"],["0","1","1"]],"card_rows":[["1"],["0","1"],["0","1","1"],["0","2","3","1"]],"row_sums_preview":["1","1","2","6","24","120","720","5040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A132393"],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":[],"edges":[],"text":"A132393"},"headline":"A132393: Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","chain":null}},{"id":"l-8cb846b7c415dc2a","kind":"fingerprint_related","node_a_id":"n-ab4410127fe29869","node_b_id":"n-5b94261d6f603245","fingerprint":"f6-34eaf45657600f7a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-34eaf45657600f7a"},"other_id":"n-ab4410127fe29869","other":{"id":"n-ab4410127fe29869","name":"reversal of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-34eaf45657600f7a","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","3","2"]],"row_sums_preview":["1","1","2","6","24","120","720","5040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["reverse"],"edges":["e-c7203dc00a9aba0b"],"text":"A132393 \u2192 reverse"},"headline":"A132393 \u2192 reverse","chain":null}},{"id":"l-c1674a480d05a212","kind":"fingerprint_related","node_a_id":"n-c2e61578a2a87aa5","node_b_id":"n-5b94261d6f603245","fingerprint":"f6-34eaf45657600f7a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-34eaf45657600f7a"},"other_id":"n-c2e61578a2a87aa5","other":{"id":"n-c2e61578a2a87aa5","name":"basis_map_A013609 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-34eaf45657600f7a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["8","12","4"]],"card_rows":[["1"],["2","2"],["8","12","4"],["48","88","48","8"]],"row_sums_preview":["1","4","24","192","1920","23040","322560","5160960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["basis A013609"],"edges":["e-cdc351e4a2e06a48"],"text":"A028338 \u2192 basis A013609"},"headline":"A028338 \u2192 basis A013609","chain":null}},{"id":"l-c6da04463acd48aa","kind":"fingerprint_related","node_a_id":"n-4d60c3dd8627c01d","node_b_id":"n-5b94261d6f603245","fingerprint":"f6-34eaf45657600f7a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-34eaf45657600f7a"},"other_id":"n-4d60c3dd8627c01d","other":{"id":"n-4d60c3dd8627c01d","name":"Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","first_index":0,"fingerprint":"f6-34eaf45657600f7a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["1","3","2"]],"card_rows":[["1"],["1","1"],["1","3","2"],["1","6","11","6"]],"row_sums_preview":["1","2","6","24","120","720","5040","40320"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A094638"],"derivation":{"root":"n-4d60c3dd8627c01d","steps":[],"edges":[],"text":"A094638"},"headline":"A094638: Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","chain":null}},{"id":"l-e5a73efd4cee36b7","kind":"fingerprint_related","node_a_id":"n-5b94261d6f603245","node_b_id":"n-f4e656360cd62235","fingerprint":"f6-34eaf45657600f7a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-34eaf45657600f7a"},"other_id":"n-f4e656360cd62235","other":{"id":"n-f4e656360cd62235","name":"basis_map_A013611 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":"f6-34eaf45657600f7a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["32","48","16"]],"card_rows":[["1"],["4","4"],["32","48","16"],["384","704","384","64"]],"row_sums_preview":["1","8","96","1536","30720","737280","20643840","660602880"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9c5c208df8605708","steps":["basis A013611"],"edges":["e-a967cf092a6717a9"],"text":"A225471 \u2192 basis A013611"},"headline":"A225471 \u2192 basis A013611","chain":null}},{"id":"l-95e7baab8886342d","kind":"oeis_match","node_a_id":"n-5b94261d6f603245","node_b_id":null,"fingerprint":null,"oeis_id":"A052690","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A052690","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ba70132dae60f907","kind":"oeis_match","node_a_id":"n-5b94261d6f603245","node_b_id":null,"fingerprint":null,"oeis_id":"A034001","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":"A034001","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A034001","relation":"match_row_sums"},{"oeis_id":"A052690","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-5b94261d6f603245","oeis_id":"A034001","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":17,"oeis_terms_available":17,"term_count":200,"raw_size":43932}],"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,           3;\n  2:         18,          27,           9;\n  3:        162,         297,         162,          27;\n  4:       1944,        4050,        2835,         810,         81;\n  5:      29160,       66582,       54675,       20655,       3645,        243;\n  6:     524880,     1285956,     1183896,      535815,     127575,      15309,       729;\n  7:   11022480,    28579716,    28719684,    14803803,    4286520,     704214,     61236,     2187;\n  8:  264539520,   718980624,   775011564,   441450324,  147287889,   29760696,   3582306,   236196,   6561;\n  9: 7142567040, 20206095408, 23082254100, 14244193440, 5301123975, 1245402459, 186004350, 17124210, 885735, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n+3*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 + 3*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 + 3*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 A013610"],"edges":["e-86f8b58a2aa07621"],"text":"A225470 \u2192 basis A013610"},"headline":"A225470 \u2192 basis A013610","chain":null}
