{"id":"n-22d90d08a08421af","revision":0,"first_index":1,"fingerprint":"f6-02d3ee6161499280","name":"basis_map_A008288 of basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-3","-2"],["9","16","4"],["-27","-82","-52","-8"],["81","352","392","144","16"],["-243","-1378","-2280","-1440","-368","-32"],["729","5104","11404","10672","4592","896","64"],["-2187","-18226","-51740","-65736","-41488","-13408","-2112","-128"],["6561","63424","219408","357216","304992","143104","36864","4864","256"],["-19683","-216514","-885712","-1773952","-1941408","-1225152","-454144","-97024","-11008","-512"],["59049","728272","3444500","8234400","11112992","8964096","4442496","1355520","247040","24576","1024"],["-177147","-2421010","-13010436","-36276168","-58644448","-58289728","-36786048","-14924288","-3860224","-612864","-54272","-2048"],["531441","7971616","48006808","153311344","290259952","345781504","268324352","138087680","47259392","10592256","1488896","118784","4096"],["-1594323","-26040610","-173781176","-626451360","-1364281488","-1906065632","-1771081728","-1117798656","-483480832","-142769664","-28203008","-3555328","-258048","-8192"],["4782969","84499120","619128732","2489447376","6145871504","9896000896","10783048128","8135743744","4303917824","1600782336","415069184","73248768","8368128","557056","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-5","29","-169","985","-5741","33461","-195025","1136689","-6625109","38613965","-225058681","1311738121","-7645370045","44560482149"],"nonn":false,"definitions":[{"position":0,"source":"derived:basis_map_A008288","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["-3","-2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","-2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":198265,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1564556493/268435456, -46056243/268435456]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"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 >= 1, 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-23f4bccf1d96c5c2"]},{"id":198266,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-23f4bccf1d96c5c2"]},{"id":198267,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-23f4bccf1d96c5c2"]}],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"incoming_edges":[{"id":"e-23f4bccf1d96c5c2","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-22d90d08a08421af","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":"89a9c98fcfea37321a75d9bf26604b262412bb6125afd497a808fdf5543759be","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"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-0793a8fd2b196f0d","role":"sequence","name":"basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-0508e890bea41fd3","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-1a789ecfc71669a3","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-1a789ecfc71669a3","other":{"id":"n-1a789ecfc71669a3","name":"basis_map_A008288 of basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3"],["27","18"],["243","432","108"]],"card_rows":[["3"],["27","18"],["243","432","108"],["2187","6642","4212","648"]],"row_sums_preview":["3","45","783","13689","239355","4185189","73179207","1279559025"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["basis A013610","basis A008288"],"edges":["e-284e6c48cfe1aa93","e-c4254293fa35d77b"],"text":"A120909 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A120909 \u2192 basis A013610 \u2192 basis A008288","chain":null}},{"id":"l-22a9277bd3d7b255","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-59aeceada33946d1","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-59aeceada33946d1","other":{"id":"n-59aeceada33946d1","name":"basis_map_A008288 of basis_map_A013609 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["9","16","4"],["27","82","52","8"],["81","352","392","144","16"]],"card_rows":[["9","16","4"],["27","82","52","8"],["81","352","392","144","16"],["243","1378","2280","1440","368","32"]],"row_sums_preview":["29","169","985","5741","33461","195025","1136689","6625109"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013609","basis A008288"],"edges":["e-7bee75fc05394bb8","e-d0ba861719d2ce0c"],"text":"basis-map coefficients of A112227 \u2192 basis A013609 \u2192 basis A008288"},"headline":"basis-map coefficients of A112227 \u2192 basis A013609 \u2192 basis A008288","chain":null}},{"id":"l-2c020bf1af5573fa","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-f522163d1450b482","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-f522163d1450b482","other":{"id":"n-f522163d1450b482","name":"basis_map_A008288 of basis_map_A013611 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","8"],["216","384","96"]],"card_rows":[["1"],["12","8"],["216","384","96"],["4320","13120","8320","1280"]],"row_sums_preview":["1","20","696","27040","1103200","46295424","1978749696","85673702400"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013611","basis A008288"],"edges":["e-6d76bdb8087cd7e0","e-73b47dbe092b389d"],"text":"A356546 \u2192 basis A013611 \u2192 basis A008288"},"headline":"A356546 \u2192 basis A013611 \u2192 basis A008288","chain":null}},{"id":"l-37764eb0c8f06e8f","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-55a69da1c48f482e","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-55a69da1c48f482e","other":{"id":"n-55a69da1c48f482e","name":"basis_map_A008288 of basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["18","32","8"]],"card_rows":[["1"],["3","2"],["18","32","8"],["162","492","312","48"]],"row_sums_preview":["1","5","58","1014","23640","688920","24091920","982926000"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013609","basis A008288"],"edges":["e-456dc355a6d78eff","e-4b40de83a7a796c9"],"text":"A136572 \u2192 basis A013609 \u2192 basis A008288"},"headline":"A136572 \u2192 basis A013609 \u2192 basis A008288","chain":null}},{"id":"l-3e0569c9dd162311","kind":"fingerprint_related","node_a_id":"n-33a50ee3b762dcdc","node_b_id":"n-22d90d08a08421af","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-33a50ee3b762dcdc","other":{"id":"n-33a50ee3b762dcdc","name":"basis_map_A008288 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["9","16","4"]],"card_rows":[["1"],["3","2"],["9","16","4"],["27","82","52","8"]],"row_sums_preview":["1","5","29","169","985","5741","33461","195025"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A008288"],"edges":["e-9c90c90150c15b0e"],"text":"A013609 \u2192 basis A008288"},"headline":"A013609 \u2192 basis A008288","chain":null}},{"id":"l-4b77036084276824","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-b3709c7e6da5a05e","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-b3709c7e6da5a05e","other":{"id":"n-b3709c7e6da5a05e","name":"basis_map_A008288 of basis_map_A013610 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["9","6"],["81","144","36"]],"card_rows":[["1"],["9","6"],["81","144","36"],["729","2214","1404","216"]],"row_sums_preview":["1","15","261","4563","79785","1395063","24393069","426519675"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A013610","basis A008288"],"edges":["e-14a3a792ab5fe5af","e-2622a4f3a668e6f9"],"text":"A013609 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A013609 \u2192 basis A013610 \u2192 basis A008288","chain":null}},{"id":"l-4cc18ff3eaa57e81","kind":"fingerprint_related","node_a_id":"n-0e4e8cd316890790","node_b_id":"n-22d90d08a08421af","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-0e4e8cd316890790","other":{"id":"n-0e4e8cd316890790","name":"reversal of basis_map_A008288 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["4","16","9"]],"card_rows":[["1"],["2","3"],["4","16","9"],["8","52","82","27"]],"row_sums_preview":["1","5","29","169","985","5741","33461","195025"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A008288","reverse"],"edges":["e-9c90c90150c15b0e","e-7fa08615432d5c61"],"text":"A013609 \u2192 basis A008288 \u2192 reverse"},"headline":"A013609 \u2192 basis A008288 \u2192 reverse","chain":null}},{"id":"l-4d7076a0653dc076","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-e9b21d7b5c1e846e","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-e9b21d7b5c1e846e","other":{"id":"n-e9b21d7b5c1e846e","name":"basis_map_A008288 of basis_map_A013611 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","8"],["216","384","96"]],"card_rows":[["1"],["12","8"],["216","384","96"],["5184","15744","9984","1536"]],"row_sums_preview":["1","20","696","32448","1891200","132272640","10793180160","1006516224000"],"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 A013611","basis A008288"],"edges":["e-413ab29272bdfecc","e-ad795f91ea37f81f"],"text":"A156992 \u2192 basis A013611 \u2192 basis A008288"},"headline":"A156992 \u2192 basis A013611 \u2192 basis A008288","chain":null}},{"id":"l-646c6295d82b58c6","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-854160786720f84e","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-854160786720f84e","other":{"id":"n-854160786720f84e","name":"basis_map_A008288 of basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["24","16"],["360","640","160"]],"card_rows":[["1"],["24","16"],["360","640","160"],["4320","13120","8320","1280"]],"row_sums_preview":["1","40","1160","27040","551600","10287872","179886336","2995584000"],"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 A013611","basis A008288"],"edges":["e-159425c642d412bb","e-9e986a799de41ab6"],"text":"A178820 \u2192 basis A013611 \u2192 basis A008288"},"headline":"A178820 \u2192 basis A013611 \u2192 basis A008288","chain":null}},{"id":"l-78f93979b15591ec","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-e9609fadb843d019","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-e9609fadb843d019","other":{"id":"n-e9609fadb843d019","name":"basis_map_A008288 of basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["9","16","4"]],"card_rows":[["1"],["3","2"],["9","16","4"],["27","82","52","8"]],"row_sums_preview":["1","5","29","169","985","5741","33461","195025"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013609","basis A008288"],"edges":["e-7869781294d5f3a8","e-926cd21e004f45e0"],"text":"basis-map coefficients of A026729 \u2192 basis A013609 \u2192 basis A008288"},"headline":"basis-map coefficients of A026729 \u2192 basis A013609 \u2192 basis A008288","chain":null}},{"id":"l-8ea0ab8a5aa20c63","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-d5b6cf9ef2f79c87","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-d5b6cf9ef2f79c87","other":{"id":"n-d5b6cf9ef2f79c87","name":"basis_map_A008288 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["18","12"],["324","576","144"]],"card_rows":[["1"],["18","12"],["324","576","144"],["5832","17712","11232","1728"]],"row_sums_preview":["1","30","1044","36504","1276560","44642016","1561156416","54594518400"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013611","basis A008288"],"edges":["e-fc7ebcb9132c171a","e-b24fa060d061029a"],"text":"A038221 \u2192 basis A013611 \u2192 basis A008288"},"headline":"A038221 \u2192 basis A013611 \u2192 basis A008288","chain":null}},{"id":"l-9415619790291e13","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-a81391ae04cc879c","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-a81391ae04cc879c","other":{"id":"n-a81391ae04cc879c","name":"basis_map_A008288 of basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["36","24"],["756","1344","336"]],"card_rows":[["1"],["36","24"],["756","1344","336"],["12096","36736","23296","3584"]],"row_sums_preview":["1","60","2436","75712","1985760","46295424","989374848","19770854400"],"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 A013611","basis A008288"],"edges":["e-377b2cf400ddb478","e-bb402982a6f21a50"],"text":"A178822 \u2192 basis A013611 \u2192 basis A008288"},"headline":"A178822 \u2192 basis A013611 \u2192 basis A008288","chain":null}},{"id":"l-950782d5cab4eea9","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-9a0b62b1efd950a7","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-9a0b62b1efd950a7","other":{"id":"n-9a0b62b1efd950a7","name":"basis_map_A008288 of basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["30","20"],["540","960","240"]],"card_rows":[["1"],["30","20"],["540","960","240"],["7560","22960","14560","2240"]],"row_sums_preview":["1","50","1740","47320","1103200","23147712","449715840","8237856000"],"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 A013611","basis A008288"],"edges":["e-4faaf8fb320b6fca","e-5aa57d53cec8b7dd"],"text":"A178821 \u2192 basis A013611 \u2192 basis A008288"},"headline":"A178821 \u2192 basis A013611 \u2192 basis A008288","chain":null}},{"id":"l-a80992d2aea09f48","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-ec43814af8c33829","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-ec43814af8c33829","other":{"id":"n-ec43814af8c33829","name":"basis_map_A008288 of basis_map_A013611 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":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["72","128","32"]],"card_rows":[["1"],["6","4"],["72","128","32"],["1080","3280","2080","320"]],"row_sums_preview":["1","10","232","6760","220640","7715904","282678528","10709212800"],"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 A013611","basis A008288"],"edges":["e-9f3d1c5f0f510549","e-e58d6370d89e244a"],"text":"A085880 \u2192 basis A013611 \u2192 basis A008288"},"headline":"A085880 \u2192 basis A013611 \u2192 basis A008288","chain":null}},{"id":"l-b1c8cd51404db4e8","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-e21c48f32e7cdb91","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-e21c48f32e7cdb91","other":{"id":"n-e21c48f32e7cdb91","name":"basis_map_A008288 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","8"],["144","256","64"]],"card_rows":[["1"],["12","8"],["144","256","64"],["1728","5248","3328","512"]],"row_sums_preview":["1","20","464","10816","252160","5878784","137056256","3195289600"],"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 A013611","basis A008288"],"edges":["e-e6fcabb7b5383fc4","e-165aef66582bb7fb"],"text":"A038208 \u2192 basis A013611 \u2192 basis A008288"},"headline":"A038208 \u2192 basis A013611 \u2192 basis A008288","chain":null}},{"id":"l-ba378e5a1f30cce3","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-b8f00920709d45e5","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-b8f00920709d45e5","other":{"id":"n-b8f00920709d45e5","name":"basis_map_A008288 of basis_map_A013611 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["12","8"],["72","128","32"]],"card_rows":[["2"],["12","8"],["72","128","32"],["432","1312","832","128"]],"row_sums_preview":["2","20","232","2704","31520","367424","4283008","49926400"],"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 A013611","basis A008288"],"edges":["e-3deac81ab612cd2e","e-a8de87894ed667b0"],"text":"A028326 \u2192 basis A013611 \u2192 basis A008288"},"headline":"A028326 \u2192 basis A013611 \u2192 basis A008288","chain":null}},{"id":"l-d35b83e07d62bfc4","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-0db0a2cf3d5aaadd","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-0db0a2cf3d5aaadd","other":{"id":"n-0db0a2cf3d5aaadd","name":"basis_map_A008288 of basis_map_A013611 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["72","128","32"]],"card_rows":[["1"],["6","4"],["72","128","32"],["1296","3936","2496","384"]],"row_sums_preview":["1","10","232","8112","378240","22045440","1541882880","125814528000"],"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 A013611","basis A008288"],"edges":["e-b8ab5069d7db191f","e-6bc33c229cb80f73"],"text":"A196347 \u2192 basis A013611 \u2192 basis A008288"},"headline":"A196347 \u2192 basis A013611 \u2192 basis A008288","chain":null}},{"id":"l-d90df6ac4222f1c2","kind":"fingerprint_related","node_a_id":"n-22d90d08a08421af","node_b_id":"n-e7fa785b1435b6e0","fingerprint":"f6-02d3ee6161499280","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02d3ee6161499280"},"other_id":"n-e7fa785b1435b6e0","other":{"id":"n-e7fa785b1435b6e0","name":"basis_map_A008288 of basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-02d3ee6161499280","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["108","192","48"]],"card_rows":[["1"],["6","4"],["108","192","48"],["3240","9840","6240","960"]],"row_sums_preview":["1","10","348","20280","1654800","173607840","22260934080","3373402032000"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013611","basis A008288"],"edges":["e-20dcbad557ba463b","e-1bb1694edf261838"],"text":"A085881 \u2192 basis A013611 \u2192 basis A008288"},"headline":"A085881 \u2192 basis A013611 \u2192 basis A008288","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:     -3,      -2;\n   3:      9,      16,       4;\n   4:    -27,     -82,     -52,       -8;\n   5:     81,     352,     392,      144,       16;\n   6:   -243,   -1378,   -2280,    -1440,     -368,      -32;\n   7:    729,    5104,   11404,    10672,     4592,      896,      64;\n   8:  -2187,  -18226,  -51740,   -65736,   -41488,   -13408,   -2112,   -128;\n   9:   6561,   63424,  219408,   357216,   304992,   143104,   36864,   4864,    256;\n  10: -19683, -216514, -885712, -1773952, -1941408, -1225152, -454144, -97024, -11008, -512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(3-2*x)*P(n-2,x) - (4+2*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = -3 - 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := -3 - 2*x;\nP[n_, x_] := P[n, x] = Expand[-(3 - 2*x)*P[n - 2, x] - (4 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -3 - 2*x; for(n = 3, N, my(i = n); v[i] = -(3 - 2*x)*v[i - 2] - (4 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013609","basis A008288"],"edges":["e-66f8525b021adc45","e-23f4bccf1d96c5c2"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013609 \u2192 basis A008288"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013609 \u2192 basis A008288","chain":null}