{"id":"n-8e5ab335b48f40f5","revision":0,"first_index":0,"fingerprint":"f6-0fee5d4c7376c95e","name":"basis_map_A008288 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","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"],["15","9"],["225","351","81"],["3375","8991","5103","729"],["50625","193671","190269","63423","6561"],["759375","3804651","5522175","3116475","728271","59049"],["11390625","70668531","138312441","114574743","44463897","7971615","531441"],["170859375","1264639311","3146307867","3519497547","1983514959","581927895","84499119","4782969"],["2562890625","22042538991","66914149653","95648928327","72421059393","30555200295","7179236469","875283327","43046721"],["38443359375","376754999571","1354557546999","2379050374923","2298192689655","1294630134075","433867900959","84845087091","8910671247","387420489"],["576650390625","6343214763051","26409378567105","55349354287575","65738125576521","47076367599531","20941756252569","5801621822775","970488324945","89494132959","3486784401"],["8649755859375","105527384281431","499948088535171","1222479367168923","1736041843749087","1525781340332775","859371092233443","314388240039099","74083708168047","10819491996303","889130022255","31381059609"],["129746337890625","1738603104516711","9243079045334397","25903210842794007","43037194287128769","45203841047338335","31165128862353405","14379774325691679","4454551871850753","912111418241991","118149689427885","8755315630911","282429536481"],["1946195068359375","28414461058740891","167605999300105983","530630594451930267","1013673154590993303","1246168219729539555","1025154973423441575","577369498487489955","224784506986022367","60271812470608587","10902250824060735","1268391048336171","85576149553743","2541865828329"],["29192926025390625","461248309565335971","2990521304572749129","10569970635564057543","22888850803392080745","32386249557131296995","31159410530124457665","20898433019019554055","9904144371420986865","3326951497655172771","785387115784776825","127158532639377111","13423593439405449","831190125863583","22876792454961"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","24","657","18198","504549","13989996","387913293","10756029042","298242333801","8269640184384","229299938502057","6358010823445518","176294428577735949","4888278175455010836","135541796263252960293"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008288","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["15","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-90","27"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["21","9"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":198370,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1564556493/268435456, -46056243/268435456]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [\u22123\u22122\u221a2, \u22123+2\u221a2] (rational outer bound [-1564556493/268435456, -46056243/268435456])","region_exact":"[\u22123\u22122\u221a2, \u22123+2\u221a2]","edges":["e-9f3cec5348d4137f"]},{"id":198371,"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-9f3cec5348d4137f"]},{"id":198372,"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-9f3cec5348d4137f"]}],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"incoming_edges":[{"id":"e-9f3cec5348d4137f","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-8e5ab335b48f40f5","premise_count":1,"theorem":"affine current multiplier with an affine initial row (module documentation, \"The affine-current theorem\"): with beta = b/d, K = beta^2 + a beta - c > 0 and eta = m/d >= 0, every real polynomial with all roots in E = [-beta - sqrt(K), -beta + sqrt(K)] has a real-rooted image, and no interval of positive length outside E works (Theorem 1); for eta > 0 the images lie in I = [(-a - 2 beta - 2 sqrt(K))/d, (-a - 2 beta + 2 sqrt(K))/d] if and only if eta >= 1/2 and (s - (2 eta - 1) beta)^2 <= (2 eta - 1)^2 K, s = l - eta a (Theorem 2); with a >= 0, K >= beta^2, a + 2 beta >= 0, a^2 + 4c >= 0, l >= 0 and the scalar criterion, PF inputs with roots in [-R, 0], R = beta + sqrt(K), have PF images, sharply (the common-interlacer proof; the Chebyshev case is Iserles and Saff, Math. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"349c94fe7f90c2d6cf2719e5d251a05ff1d2c29282e33032b64b8778ef8d5f0c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-9dc2047d1a16a51a","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-02ee689480fba3b2","kind":"fingerprint_related","node_a_id":"n-f5630447d9b46bac","node_b_id":"n-8e5ab335b48f40f5","fingerprint":"f6-0fee5d4c7376c95e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0fee5d4c7376c95e"},"other_id":"n-f5630447d9b46bac","other":{"id":"n-f5630447d9b46bac","name":"basis_map_A008288 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-0fee5d4c7376c95e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","3"],["25","39","9"]],"card_rows":[["1"],["5","3"],["25","39","9"],["125","333","189","27"]],"row_sums_preview":["1","8","73","674","6229","57572","532117","4918166"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-5733d79f45a14084","steps":["basis A008288"],"edges":["e-28556ceb40718bbf"],"text":"A013620 \u2192 basis A008288"},"headline":"A013620 \u2192 basis A008288","chain":null}},{"id":"l-03c99c75dedae7db","kind":"fingerprint_related","node_a_id":"n-75d573c0ca4216ce","node_b_id":"n-8e5ab335b48f40f5","fingerprint":"f6-0fee5d4c7376c95e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0fee5d4c7376c95e"},"other_id":"n-75d573c0ca4216ce","other":{"id":"n-75d573c0ca4216ce","name":"basis_map_A008288 of basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-0fee5d4c7376c95e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["30","18"],["525","819","189"]],"card_rows":[["1"],["30","18"],["525","819","189"],["7000","18648","10584","1512"]],"row_sums_preview":["1","48","1533","37744","784854","14508144","245838054","3895187472"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013610","basis A008288"],"edges":["e-3fa86ece9d0bedd1","e-31efb735b14ec404"],"text":"A178822 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A178822 \u2192 basis A013610 \u2192 basis A008288","chain":null}},{"id":"l-11d974fa42a895ec","kind":"fingerprint_related","node_a_id":"n-7779a9e47344358a","node_b_id":"n-8e5ab335b48f40f5","fingerprint":"f6-0fee5d4c7376c95e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0fee5d4c7376c95e"},"other_id":"n-7779a9e47344358a","other":{"id":"n-7779a9e47344358a","name":"basis_map_A008288 of basis_map_A013610 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-0fee5d4c7376c95e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["10","6"],["50","78","18"]],"card_rows":[["2"],["10","6"],["50","78","18"],["250","666","378","54"]],"row_sums_preview":["2","16","146","1348","12458","115144","1064234","9836332"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013610","basis A008288"],"edges":["e-28b777b3c0b25332","e-9556e50b57d29665"],"text":"A028326 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A028326 \u2192 basis A013610 \u2192 basis A008288","chain":null}},{"id":"l-67304a9f6f5331ee","kind":"fingerprint_related","node_a_id":"n-a6a2221751b3af6f","node_b_id":"n-8e5ab335b48f40f5","fingerprint":"f6-0fee5d4c7376c95e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0fee5d4c7376c95e"},"other_id":"n-a6a2221751b3af6f","other":{"id":"n-a6a2221751b3af6f","name":"basis_map_A008288 of basis_map_A013609 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-0fee5d4c7376c95e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["10","6"],["100","156","36"]],"card_rows":[["1"],["10","6"],["100","156","36"],["1000","2664","1512","216"]],"row_sums_preview":["1","16","292","5392","99664","1842304","34055488","629525248"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A013610","basis A008288"],"edges":["e-1a1a6d136d357a71","e-a6b67005f22343a4"],"text":"A038208 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A038208 \u2192 basis A013610 \u2192 basis A008288","chain":null}},{"id":"l-8477c9fbd567491d","kind":"fingerprint_related","node_a_id":"n-8e5ab335b48f40f5","node_b_id":"n-056fa906003b296d","fingerprint":"f6-0fee5d4c7376c95e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0fee5d4c7376c95e"},"other_id":"n-056fa906003b296d","other":{"id":"n-056fa906003b296d","name":"basis_map_A008288 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-0fee5d4c7376c95e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3600","5616","1296"],["216000","575424","326592","46656"],["12960000","49579776","48708864","16236288","1679616"]],"card_rows":[["3600","5616","1296"],["216000","575424","326592","46656"],["12960000","49579776","48708864","16236288","1679616"],["777600000","3895962624","5654707200","3191270400","745749504","60466176"]],"row_sums_preview":["10512","1164672","129164544","14325755904","1588892848128","176226779824128","19545609587982336","2167836556495159296"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013610","basis A008288"],"edges":["e-85e8843cd1dca083","e-d120a6d023071b51"],"text":"A038338 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A038338 \u2192 basis A013610 \u2192 basis A008288","chain":null}},{"id":"l-aa604b5b5cae50cc","kind":"fingerprint_related","node_a_id":"n-96a7112290a52677","node_b_id":"n-8e5ab335b48f40f5","fingerprint":"f6-0fee5d4c7376c95e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0fee5d4c7376c95e"},"other_id":"n-96a7112290a52677","other":{"id":"n-96a7112290a52677","name":"basis_map_A008288 of basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-0fee5d4c7376c95e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["25","15"],["375","585","135"]],"card_rows":[["1"],["25","15"],["375","585","135"],["4375","11655","6615","945"]],"row_sums_preview":["1","40","1095","23590","436030","7254072","111744570","1622994780"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013610","basis A008288"],"edges":["e-0c010b1c5e20b0cb","e-1abdf6dc0212b840"],"text":"A178821 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A178821 \u2192 basis A013610 \u2192 basis A008288","chain":null}},{"id":"l-b359a4c833ff8215","kind":"fingerprint_related","node_a_id":"n-8e5ab335b48f40f5","node_b_id":"n-8d3044b53af1a6c6","fingerprint":"f6-0fee5d4c7376c95e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0fee5d4c7376c95e"},"other_id":"n-8d3044b53af1a6c6","other":{"id":"n-8d3044b53af1a6c6","name":"basis_map_A008288 of basis_map_A013610 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-0fee5d4c7376c95e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["10","6"],["150","234","54"]],"card_rows":[["1"],["10","6"],["150","234","54"],["3000","7992","4536","648"]],"row_sums_preview":["1","16","438","16176","747480","41451840","2681869680","198300453120"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013610","basis A008288"],"edges":["e-1eb90284df2d6b2d","e-e275894882451d3f"],"text":"A156992 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A156992 \u2192 basis A013610 \u2192 basis A008288","chain":null}},{"id":"l-ced67d15948f2f23","kind":"fingerprint_related","node_a_id":"n-9dd73c3d712d6411","node_b_id":"n-8e5ab335b48f40f5","fingerprint":"f6-0fee5d4c7376c95e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0fee5d4c7376c95e"},"other_id":"n-9dd73c3d712d6411","other":{"id":"n-9dd73c3d712d6411","name":"basis_map_A008288 of basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-0fee5d4c7376c95e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["20","12"],["250","390","90"]],"card_rows":[["1"],["20","12"],["250","390","90"],["2500","6660","3780","540"]],"row_sums_preview":["1","32","730","13480","218015","3224032","44697828","590179920"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013610","basis A008288"],"edges":["e-24fdbe8219e7135e","e-ee6e9748ff31aac7"],"text":"A178820 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A178820 \u2192 basis A013610 \u2192 basis A008288","chain":null}},{"id":"l-e3f28b80bc7f9a6a","kind":"fingerprint_related","node_a_id":"n-7436bc4920832f70","node_b_id":"n-8e5ab335b48f40f5","fingerprint":"f6-0fee5d4c7376c95e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0fee5d4c7376c95e"},"other_id":"n-7436bc4920832f70","other":{"id":"n-7436bc4920832f70","name":"basis_map_A008288 of basis_map_A013610 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-0fee5d4c7376c95e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","3"],["50","78","18"]],"card_rows":[["1"],["5","3"],["50","78","18"],["625","1665","945","135"]],"row_sums_preview":["1","8","146","3370","87206","2418024","70239444","2109893214"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013610","basis A008288"],"edges":["e-13e67b7b83b67a66","e-0c5eb7e17f6b9f42"],"text":"A085880 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A085880 \u2192 basis A013610 \u2192 basis A008288","chain":null}},{"id":"l-f92d9605ec067ce6","kind":"fingerprint_related","node_a_id":"n-7b22e4ad64916eb8","node_b_id":"n-8e5ab335b48f40f5","fingerprint":"f6-0fee5d4c7376c95e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0fee5d4c7376c95e"},"other_id":"n-7b22e4ad64916eb8","other":{"id":"n-7b22e4ad64916eb8","name":"reversal of basis_map_A008288 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-0fee5d4c7376c95e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","5"],["9","39","25"]],"card_rows":[["1"],["3","5"],["9","39","25"],["27","189","333","125"]],"row_sums_preview":["1","8","73","674","6229","57572","532117","4918166"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5733d79f45a14084","steps":["basis A008288","reverse"],"edges":["e-28556ceb40718bbf","e-9a4f8de1e11ee45a"],"text":"A013620 \u2192 basis A008288 \u2192 reverse"},"headline":"A013620 \u2192 basis A008288 \u2192 reverse","chain":null}},{"id":"l-ff0d603bab5418aa","kind":"fingerprint_related","node_a_id":"n-8e5ab335b48f40f5","node_b_id":"n-4a08a6e883134505","fingerprint":"f6-0fee5d4c7376c95e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0fee5d4c7376c95e"},"other_id":"n-4a08a6e883134505","other":{"id":"n-4a08a6e883134505","name":"basis_map_A008288 of basis_map_A013610 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-0fee5d4c7376c95e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","3"],["50","78","18"]],"card_rows":[["1"],["5","3"],["50","78","18"],["750","1998","1134","162"]],"row_sums_preview":["1","8","146","4044","149496","6908640","383124240","24787556640"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013610","basis A008288"],"edges":["e-542183e09dec6678","e-0dfe057e4c16e7ca"],"text":"A196347 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A196347 \u2192 basis A013610 \u2192 basis A008288","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:          15,            9;\n  2:         225,          351,            81;\n  3:        3375,         8991,          5103,           729;\n  4:       50625,       193671,        190269,         63423,          6561;\n  5:      759375,      3804651,       5522175,       3116475,        728271,         59049;\n  6:    11390625,     70668531,     138312441,     114574743,      44463897,       7971615,       531441;\n  7:   170859375,   1264639311,    3146307867,    3519497547,    1983514959,     581927895,     84499119,     4782969;\n  8:  2562890625,  22042538991,   66914149653,   95648928327,   72421059393,   30555200295,   7179236469,   875283327,   43046721;\n  9: 38443359375, 376754999571, 1354557546999, 2379050374923, 2298192689655, 1294630134075, 433867900959, 84845087091, 8910671247, 387420489;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(90-27*x)*P(n-2,x) + (21+9*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 15 + 9*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 15 + 9*x;\nP[n_, x_] := P[n, x] = Expand[-(90 - 27*x)*P[n - 2, x] + (21 + 9*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] = 15 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = -(90 - 27*x)*v[i - 2] + (21 + 9*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-8d88ef9de74b305f","steps":["basis A013610","basis A008288"],"edges":["e-b88e4ed4246b1dea","e-9f3cec5348d4137f"],"text":"A038221 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A038221 \u2192 basis A013610 \u2192 basis A008288","chain":null}