{"id":"n-29f388dd4604fb27","revision":1,"first_index":0,"fingerprint":"f6-b5f1389370a379f1","name":"basis_map_A008277 of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","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"],["4","2"],["8","10","2"],["32","76","36","4"],["64","260","220","56","4"],["256","1688","2280","1000","160","8"],["512","5320","10808","7280","1960","216","8"],["2048","32944","97104","94416","38080","6944","560","16"],["4096","100880","421360","569184","322896","86688","11424","704","16"],["16384","613472","3572640","6534560","5013120","1859424","356160","35520","1728","32"],["32768","1856800","14904032","36245440","36613280","18029088","4708704","675840","52800","2080","32"],["131072","11206336","122945856","392262464","511850240","325633792","111397440","21582528","2407680","151360","4928","64"],["262144","33750080","502989760","2084258176","3463363904","2791286784","1216813312","305640192","45659328","4072640","210496","5760","64"],["1048576","203024768","4091418240","21848561280","45728883200","46004742784","25051586560","7935150080","1524466944","180916736","13197184","570752","13312","128"],["2097152","610122880","16568697728","113334224640","296221860480","367762082688","246417435264","96467937280","23179819520","3514551040","339282944","20616960","757120","15232","128"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","20","148","604","5392","26104","272112","1517248","18003040","113120864","1499573760","10448312640","152583580544","1164439501056"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["8","10","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","10","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","2"]]},"deriv_order":2,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162264,"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-f15543b521e0834d"]},{"id":162265,"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-f15543b521e0834d"]},{"id":162266,"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-f15543b521e0834d"]}],"region_exact":{},"incoming_edges":[{"id":"e-f15543b521e0834d","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-29f388dd4604fb27","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"a36b0b7e6fff566ce528f6d46dc67e9a44ac98eebc9f29220c43ec522d87d6f4","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-a624e5aba8e1c39e","role":"sequence","name":"Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","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-0c4e0669921bad87","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-34783469fb5242d8","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-34783469fb5242d8","other":{"id":"n-34783469fb5242d8","name":"basis_map_A008277 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["8","4"],["40","50","10"]],"card_rows":[["1"],["8","4"],["40","50","10"],["160","380","180","20"]],"row_sums_preview":["1","12","100","740","5285","37744","274092","2040840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A008277"],"edges":["e-96de39e5a8006173"],"text":"A178820 \u2192 basis A008277"},"headline":"A178820 \u2192 basis A008277","chain":null}},{"id":"l-13d802c6fc3d2476","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-272049576da882c4","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-272049576da882c4","other":{"id":"n-272049576da882c4","name":"basis_map_A008277 of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3"],["30","15"],["180","225","45"]],"card_rows":[["3"],["30","15"],["180","225","45"],["840","1995","945","105"]],"row_sums_preview":["3","45","450","3885","31710","254772","2055690","16836930"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["basis A008277"],"edges":["e-19677b98e028e07a"],"text":"A344565 \u2192 basis A008277"},"headline":"A344565 \u2192 basis A008277","chain":null}},{"id":"l-1837658bc8f8677e","kind":"fingerprint_related","node_a_id":"n-0990b5be9f344750","node_b_id":"n-29f388dd4604fb27","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-0990b5be9f344750","other":{"id":"n-0990b5be9f344750","name":"Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","4"]],"card_rows":[["1"],["1","2"],["1","5","4"],["1","9","19","8"]],"row_sums_preview":["1","3","10","37","151","674","3263","17007"],"regions":["(-inf, 0)"],"region_exact":{},"oeis_ids":["A126351"],"derivation":{"root":"n-0990b5be9f344750","steps":[],"edges":[],"text":"A126351"},"headline":"A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.","chain":null}},{"id":"l-21da66d63bf62dc0","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-11f206dfc8a798d6","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-11f206dfc8a798d6","other":{"id":"n-11f206dfc8a798d6","name":"basis_map_A008277 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["12","15","3"]],"card_rows":[["1"],["2","1"],["12","15","3"],["120","285","135","15"]],"row_sums_preview":["1","3","30","555","15855","636930","33918885","2298240945"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A008277"],"edges":["e-380a606cb73e0af7"],"text":"A085881 \u2192 basis A008277"},"headline":"A085881 \u2192 basis A008277","chain":null}},{"id":"l-2c3f3e8f2c068cba","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-d06008b4ada4e713","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-d06008b4ada4e713","other":{"id":"n-d06008b4ada4e713","name":"basis_map_A008277 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["8","10","2"]],"card_rows":[["1"],["2","1"],["8","10","2"],["48","114","54","6"]],"row_sums_preview":["1","3","20","222","3624","80880","2349360","85715280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A008277"],"edges":["e-73c89ff38d5bbc45"],"text":"A196347 \u2192 basis A008277"},"headline":"A196347 \u2192 basis A008277","chain":null}},{"id":"l-34e5a56aa2cc8bc9","kind":"fingerprint_related","node_a_id":"n-6aaec68d653cbdb0","node_b_id":"n-29f388dd4604fb27","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-6aaec68d653cbdb0","other":{"id":"n-6aaec68d653cbdb0","name":"Triangle read by rows: 2-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["4","5","1"]],"card_rows":[["1"],["2","1"],["4","5","1"],["8","19","9","1"]],"row_sums_preview":["1","3","10","37","151","674","3263","17007"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A143494"],"derivation":{"root":"n-6aaec68d653cbdb0","steps":[],"edges":[],"text":"A143494"},"headline":"A143494: Triangle read by rows: 2-Stirling numbers of the second kind.","chain":null}},{"id":"l-4a5cf38d3532006d","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-085c06a40df81890","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-085c06a40df81890","other":{"id":"n-085c06a40df81890","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["576","720","144"],["13824","32832","15552","1728"],["331776","1347840","1140480","290304","20736"]],"card_rows":[["576","720","144"],["13824","32832","15552","1728"],["331776","1347840","1140480","290304","20736"],["7962624","52503552","70917120","31104000","4976640","248832"]],"row_sums_preview":["1440","63936","3131136","167712768","9743265792","609391558656","40774304268288","2902866627133440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A008277"],"edges":["e-44f842c6e5783c9b"],"text":"A038338 \u2192 basis A008277"},"headline":"A038338 \u2192 basis A008277","chain":null}},{"id":"l-5c9cf3742cb5e040","kind":"fingerprint_related","node_a_id":"n-a514cae29d6b9cc2","node_b_id":"n-29f388dd4604fb27","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-a514cae29d6b9cc2","other":{"id":"n-a514cae29d6b9cc2","name":"basis_map_A008277 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-2","-1"],["4","5","1"]],"card_rows":[["1"],["-2","-1"],["4","5","1"],["-8","-19","-9","-1"]],"row_sums_preview":["1","-3","10","-37","151","-674","3263","-17007"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["basis A008277"],"edges":["e-0bd7ea1430928905"],"text":"basis-map coefficients of A113214, shifted \u2192 basis A008277"},"headline":"basis-map coefficients of A113214, shifted \u2192 basis A008277","chain":null}},{"id":"l-60adc48b445188c3","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-c990ad92ffd076e4","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-c990ad92ffd076e4","other":{"id":"n-c990ad92ffd076e4","name":"reversal of Triangle read by rows: 2-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","4"]],"card_rows":[["1"],["1","2"],["1","5","4"],["1","9","19","8"]],"row_sums_preview":["1","3","10","37","151","674","3263","17007"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6aaec68d653cbdb0","steps":["reverse"],"edges":["e-e104c7d9553deadb"],"text":"A143494 \u2192 reverse"},"headline":"A143494 \u2192 reverse","chain":null}},{"id":"l-7acf0f05cead61d8","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-50766ce7e49984e9","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-50766ce7e49984e9","other":{"id":"n-50766ce7e49984e9","name":"basis_map_A008277 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["24","30","6"]],"card_rows":[["1"],["4","2"],["24","30","6"],["192","456","216","24"]],"row_sums_preview":["1","6","60","888","18120","485280","16445520","685722240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A008277"],"edges":["e-d06f83defd222c3e"],"text":"A156992 \u2192 basis A008277"},"headline":"A156992 \u2192 basis A008277","chain":null}},{"id":"l-7f0ee8324fb52ca3","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-b1fe43ecd25295ad","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-b1fe43ecd25295ad","other":{"id":"n-b1fe43ecd25295ad","name":"basis_map_A008277 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["24","30","6"]],"card_rows":[["1"],["4","2"],["24","30","6"],["160","380","180","20"]],"row_sums_preview":["1","6","60","740","10570","169848","3015012","58368024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A008277"],"edges":["e-398f569c764d9a1d"],"text":"A356546 \u2192 basis A008277"},"headline":"A356546 \u2192 basis A008277","chain":null}},{"id":"l-85d1dd44e241ecf4","kind":"fingerprint_related","node_a_id":"n-005d2be3e1674578","node_b_id":"n-29f388dd4604fb27","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-005d2be3e1674578","other":{"id":"n-005d2be3e1674578","name":"basis_map_A008277 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["4","2"],["8","10","2"]],"card_rows":[["2"],["4","2"],["8","10","2"],["16","38","18","2"]],"row_sums_preview":["2","6","20","74","302","1348","6526","34014"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A008277"],"edges":["e-fac469530424e761"],"text":"A028326 \u2192 basis A008277"},"headline":"A028326 \u2192 basis A008277","chain":null}},{"id":"l-88a88a34a4635e30","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-9b47026307c031f7","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-9b47026307c031f7","other":{"id":"n-9b47026307c031f7","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","3"],["36","45","9"]],"card_rows":[["1"],["6","3"],["36","45","9"],["216","513","243","27"]],"row_sums_preview":["1","9","90","999","12231","163782","2378727","37194309"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A008277"],"edges":["e-d86682f478f8bbfa"],"text":"A038221 \u2192 basis A008277"},"headline":"A038221 \u2192 basis A008277","chain":null}},{"id":"l-9a7653b24eae53db","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-c8044fd3d7ed3bae","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-c8044fd3d7ed3bae","other":{"id":"n-c8044fd3d7ed3bae","name":"basis_map_A008277 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["10","5"],["60","75","15"]],"card_rows":[["1"],["10","5"],["60","75","15"],["280","665","315","35"]],"row_sums_preview":["1","15","150","1295","10570","84924","685230","5612310"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A008277"],"edges":["e-2482ec2370e1f9d0"],"text":"A178821 \u2192 basis A008277"},"headline":"A178821 \u2192 basis A008277","chain":null}},{"id":"l-9de53d06bd3c5fb3","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-c072d889dfa1d499","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-c072d889dfa1d499","other":{"id":"n-c072d889dfa1d499","name":"basis_map_A008277 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","6"],["84","105","21"]],"card_rows":[["1"],["12","6"],["84","105","21"],["448","1064","504","56"]],"row_sums_preview":["1","18","210","2072","19026","169848","1507506","13469544"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A008277"],"edges":["e-af56d17dbcb6f76e"],"text":"A178822 \u2192 basis A008277"},"headline":"A178822 \u2192 basis A008277","chain":null}},{"id":"l-a12b17953b6fe58a","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-873e284fbdbad5b5","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-873e284fbdbad5b5","other":{"id":"n-873e284fbdbad5b5","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["16","20","4"]],"card_rows":[["1"],["4","2"],["16","20","4"],["64","152","72","8"]],"row_sums_preview":["1","6","40","296","2416","21568","208832","2176896"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A008277"],"edges":["e-21aff6157efb6757"],"text":"A038208 \u2192 basis A008277"},"headline":"A038208 \u2192 basis A008277","chain":null}},{"id":"l-b8391b4f2d9fb8ff","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-646faea170766425","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-646faea170766425","other":{"id":"n-646faea170766425","name":"basis_map_A008277 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["22","11"],["484","605","121"]],"card_rows":[["1"],["22","11"],["484","605","121"],["10648","25289","11979","1331"]],"row_sums_preview":["1","33","1210","49247","2210791","108548374","5780603543","331418317197"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A008277"],"edges":["e-f221360d3c3c712f"],"text":"A038325 \u2192 basis A008277"},"headline":"A038325 \u2192 basis A008277","chain":null}},{"id":"l-cc50ff5b1290a153","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-d7033d5397b0761f","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-d7033d5397b0761f","other":{"id":"n-d7033d5397b0761f","name":"basis_map_A008277 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-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["8","10","2"]],"card_rows":[["1"],["2","1"],["8","10","2"],["40","95","45","5"]],"row_sums_preview":["1","3","20","185","2114","28308","430716","7296003"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A008277"],"edges":["e-63a786f336006cd7"],"text":"A085880 \u2192 basis A008277"},"headline":"A085880 \u2192 basis A008277","chain":null}},{"id":"l-e6ca3a31056cdeda","kind":"fingerprint_related","node_a_id":"n-c514ba2a1328f4f7","node_b_id":"n-29f388dd4604fb27","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-c514ba2a1328f4f7","other":{"id":"n-c514ba2a1328f4f7","name":"basis_map_A008277 of binomial","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["4","5","1"]],"card_rows":[["1"],["2","1"],["4","5","1"],["8","19","9","1"]],"row_sums_preview":["1","3","10","37","151","674","3263","17007"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0990b5be9f344750","steps":["reverse"],"edges":["e-a499c43746a8b06c"],"text":"A126351 \u2192 reverse"},"headline":"A126351 \u2192 reverse","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:     4,      2;\n  2:     8,     10,       2;\n  3:    32,     76,      36,       4;\n  4:    64,    260,     220,      56,       4;\n  5:   256,   1688,    2280,    1000,     160,       8;\n  6:   512,   5320,   10808,    7280,    1960,     216,      8;\n  7:  2048,  32944,   97104,   94416,   38080,    6944,    560,    16;\n  8:  4096, 100880,  421360,  569184,  322896,   86688,  11424,   704,   16;\n  9: 16384, 613472, 3572640, 6534560, 5013120, 1859424, 356160, 35520, 1728, 32;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+10*x+2*x^2)*P(n-2,x) + (10*x+4*x^2)*d/dx P(n-2,x) + 2*x^2*d^2/dx^2 P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(8 + 10*x + 2*x^2)*P[n - 2, x] + (10*x + 4*x^2)*D[P[n - 2, x], x] + 2*x^2*D[P[n - 2, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 10*x + 2*x^2)*v[i - 2] + (10*x + 4*x^2)*deriv(v[i - 2], x) + 2*x^2*deriv(deriv(v[i - 2], x), x)); 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-a624e5aba8e1c39e","steps":["basis A008277"],"edges":["e-f15543b521e0834d"],"text":"A135838 \u2192 basis A008277"},"headline":"A135838 \u2192 basis A008277","chain":null}