{"id":"n-a878b3f1264e21ba","revision":0,"first_index":0,"fingerprint":"f6-dd8e883ba3f13bd7","name":"basis_map_A013610 of 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.","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"],["8","12"],["96","240","144"],["1536","4992","5184","1728"],["30720","118272","163584","96768","20736"],["737280","3207168","5345280","4285440","1658880","248832"],["20643840","98648064","188153856","184135680","97873920","26873856","2985984"],["660602880","3404464128","7204700160","8150188032","5341593600","2034450432","418037760","35831808"],["23781703680","130487943168","300222775296","379863171072","290099625984","137339338752","39462764544","6306398208","429981696"],["951268147200","5504898170880","13574766329856","18797200146432","16162343092224","8974769061888","3226582646784","725809102848","92876046336","5159780352"],["41855798476800","253630737285120","663348496564224","989974002401280","936709497815040","588837955829760","249666865201152","70654592286720","12796255272960","1341542891520","61917364224"],["2009078326886400","12676544971407360","34884296682504192","55478934074032128","56841743923937280","39504735853608960","19050064999612416","6387422812176384","1462075360542720","217949122068480","19070548180992","743008370688"],["104472072998092800","683289278435819520","1966101967147106304","3303516132039720960","3621517892933124096","2736347191474913280","1464660210223153152","560746766228520960","152676992494338048","28878258674073600","3607057970233344","267483013447680","8916100448256"],["5850436087893196800","39517864468383006720","118301181501467787264","208590126999989649408","242447195588731600896","196693657437792632832","114857138070195535872","48977741431475011584","15278872774425182208","3449306395680178176","548534350421950464","58263744395870208","3709097786474496","106993205379072"],["351026165273591808000","2441277101157698764800","7572285263708663316480","13935021798016992411648","17049913259323771846656","14710985793332337180672","9251752173465243746304","4316950142730847125504","1504465263643211071488","390304857033912877056","74303737773479165952","10078236868815618048","921710799938912256","50928765760438272","1283918464548864"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","20","480","13440","430080","15482880","619315200","27249868800","1307993702400","68015672524800","3808877661388800","228532659683328000","14626090219732992000","994574134941843456000","71609337715812728832000"],"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":[["4","12"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203683,"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-82b0c00a778d243a"]},{"id":203684,"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-82b0c00a778d243a"]},{"id":203685,"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-82b0c00a778d243a"]},{"id":203686,"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-82b0c00a778d243a"]}],"region_exact":{},"incoming_edges":[{"id":"e-82b0c00a778d243a","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-a878b3f1264e21ba","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":"74852305967fff2fe2817579d1ef25dc070a2c441003dc91eebec53936552cc3","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-f4e656360cd62235","role":"sequence","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.","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-7523d7a7172e7feb","kind":"fingerprint_related","node_a_id":"n-d5fa840455d1e647","node_b_id":"n-a878b3f1264e21ba","fingerprint":"f6-dd8e883ba3f13bd7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd8e883ba3f13bd7"},"other_id":"n-d5fa840455d1e647","other":{"id":"n-d5fa840455d1e647","name":"basis_map_A013610 of 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-dd8e883ba3f13bd7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["24","60","36"]],"card_rows":[["1"],["4","6"],["24","60","36"],["192","624","648","216"]],"row_sums_preview":["1","10","120","1680","26880","483840","9676800","212889600"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["basis A013609","basis A013610"],"edges":["e-03658933e548767a","e-4caeccb95115fd31"],"text":"A028338 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A028338 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-97f162b719bc83da","kind":"fingerprint_related","node_a_id":"n-a6f7ad8574b034ac","node_b_id":"n-a878b3f1264e21ba","fingerprint":"f6-dd8e883ba3f13bd7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd8e883ba3f13bd7"},"other_id":"n-a6f7ad8574b034ac","other":{"id":"n-a6f7ad8574b034ac","name":"basis_map_A013609 of basis_map_A013610 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-dd8e883ba3f13bd7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["72","180","108"]],"card_rows":[["1"],["4","6"],["72","180","108"],["2880","9360","9720","3240"]],"row_sums_preview":["1","10","360","25200","2822400","457228800","100590336000","28768836096000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1336d058dca9979e","steps":["basis A013610","basis A013609"],"edges":["e-05481f5802c9cdfa","e-9c3505dcd599244b"],"text":"A204420 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A204420 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-a71cb50c9a17b64f","kind":"fingerprint_related","node_a_id":"n-c89ee0d57229b178","node_b_id":"n-a878b3f1264e21ba","fingerprint":"f6-dd8e883ba3f13bd7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd8e883ba3f13bd7"},"other_id":"n-c89ee0d57229b178","other":{"id":"n-c89ee0d57229b178","name":"basis_map_A013610 of 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-dd8e883ba3f13bd7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["6","15","9"]],"card_rows":[["1"],["2","3"],["6","15","9"],["24","78","81","27"]],"row_sums_preview":["1","5","30","210","1680","15120","151200","1663200"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["basis A013610"],"edges":["e-4deb1a58186d6d0a"],"text":"A130534 \u2192 basis A013610"},"headline":"A130534 \u2192 basis A013610","chain":null}},{"id":"l-ae6ec953ae310b3f","kind":"fingerprint_related","node_a_id":"n-3141285c8fc77c60","node_b_id":"n-a878b3f1264e21ba","fingerprint":"f6-dd8e883ba3f13bd7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd8e883ba3f13bd7"},"other_id":"n-3141285c8fc77c60","other":{"id":"n-3141285c8fc77c60","name":"reversal of basis_map_A013610 of 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-dd8e883ba3f13bd7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["9","15","6"]],"card_rows":[["1"],["3","2"],["9","15","6"],["27","81","78","24"]],"row_sums_preview":["1","5","30","210","1680","15120","151200","1663200"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["basis A013610","reverse"],"edges":["e-4deb1a58186d6d0a","e-42c6c53bca30515e"],"text":"A130534 \u2192 basis A013610 \u2192 reverse"},"headline":"A130534 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-b6828dce4b11ae58","kind":"fingerprint_related","node_a_id":"n-f6ce73d04c355dca","node_b_id":"n-a878b3f1264e21ba","fingerprint":"f6-dd8e883ba3f13bd7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd8e883ba3f13bd7"},"other_id":"n-f6ce73d04c355dca","other":{"id":"n-f6ce73d04c355dca","name":"basis_map_A013610 of 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.","first_index":0,"fingerprint":"f6-dd8e883ba3f13bd7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","9"],["54","135","81"]],"card_rows":[["1"],["6","9"],["54","135","81"],["648","2106","2187","729"]],"row_sums_preview":["1","15","270","5670","136080","3674160","110224800","3637418400"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8229dbdd6a4b5cd9","steps":["basis A013610","basis A013610"],"edges":["e-86f8b58a2aa07621","e-02a40a7a2efe22d0"],"text":"A225470 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A225470 \u2192 basis A013610 \u2192 basis A013610","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:            8,            12;\n  2:           96,           240,            144;\n  3:         1536,          4992,           5184,           1728;\n  4:        30720,        118272,         163584,          96768,          20736;\n  5:       737280,       3207168,        5345280,        4285440,        1658880,        248832;\n  6:     20643840,      98648064,      188153856,      184135680,       97873920,      26873856,       2985984;\n  7:    660602880,    3404464128,     7204700160,     8150188032,     5341593600,    2034450432,     418037760,     35831808;\n  8:  23781703680,  130487943168,   300222775296,   379863171072,   290099625984,  137339338752,   39462764544,   6306398208,   429981696;\n  9: 951268147200, 5504898170880, 13574766329856, 18797200146432, 16162343092224, 8974769061888, 3226582646784, 725809102848, 92876046336, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+4+12*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 + 4 + 12*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 + 4 + 12*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 A013611","basis A013610"],"edges":["e-8a4919ac2dd1c88f","e-82b0c00a778d243a"],"text":"A225471 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A225471 \u2192 basis A013611 \u2192 basis A013610","chain":null}
