{"id":"n-1e5aa25413f55f87","revision":1,"first_index":1,"fingerprint":"f6-03d7b8417365c509","name":"basis_map_A013611 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["1","4"],["1","8","16"],["1","12","48","64"],["1","16","96","256","256"],["1","20","160","640","1280","1024"],["1","24","240","1280","3840","6144","4096"],["1","28","336","2240","8960","21504","28672","16384"],["1","32","448","3584","17920","57344","114688","131072","65536"],["1","36","576","5376","32256","129024","344064","589824","589824","262144"],["1","40","720","7680","53760","258048","860160","1966080","2949120","2621440","1048576"],["1","44","880","10560","84480","473088","1892352","5406720","10813440","14417920","11534336","4194304"],["1","48","1056","14080","126720","811008","3784704","12976128","32440320","57671680","69206016","50331648","16777216"],["1","52","1248","18304","183040","1317888","7028736","28114944","84344832","187432960","299892736","327155712","218103808","67108864"],["1","56","1456","23296","256256","2050048","12300288","56229888","196804608","524812288","1049624576","1526726656","1526726656","939524096","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","25","125","625","3125","15625","78125","390625","1953125","9765625","48828125","244140625","1220703125","6103515625"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167609,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-f93ad378f9476e41"]},{"id":167610,"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-f93ad378f9476e41"]},{"id":167611,"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-f93ad378f9476e41"]}],"region_exact":{},"incoming_edges":[{"id":"e-f93ad378f9476e41","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-1e5aa25413f55f87","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":"44fe4fedecb5428e98931b0205ea743e9e2aeb39c1196459c4d8ef2640c20cca","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-0fa654ac9b1d20af","kind":"fingerprint_related","node_a_id":"n-ef5c9478e7d83dab","node_b_id":"n-1e5aa25413f55f87","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-ef5c9478e7d83dab","other":{"id":"n-ef5c9478e7d83dab","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["8","2"]],"card_rows":[["1"],["8","2"],["48","24","3"],["256","192","48","4"]],"row_sums_preview":["0","1","10","75","500","3125","18750","109375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dcaebc50e901d48","steps":["D"],"edges":["e-62aaaa59f91d42c1"],"text":"A038231 \u2192 D"},"headline":"A038231 \u2192 D","chain":null}},{"id":"l-209720a83de15832","kind":"fingerprint_related","node_a_id":"n-1e5aa25413f55f87","node_b_id":"n-21b46279e9abe0a4","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-21b46279e9abe0a4","other":{"id":"n-21b46279e9abe0a4","name":"basis_map_A013611 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","4"],["256","128","16"]],"card_rows":[["1"],["16","4"],["256","128","16"],["4096","3072","768","64"]],"row_sums_preview":["1","20","400","8000","160000","3200000","64000000","1280000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013611"],"edges":["e-9c08db68889738c4"],"text":"A027467 \u2192 basis A013611"},"headline":"A027467 \u2192 basis A013611","chain":null}},{"id":"l-34447c53f5d83851","kind":"fingerprint_related","node_a_id":"n-1e5aa25413f55f87","node_b_id":"n-8547d44959e7b9a0","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-8547d44959e7b9a0","other":{"id":"n-8547d44959e7b9a0","name":"basis_map_A013611 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-4"],["1","8","16"]],"card_rows":[["1"],["-1","-4"],["1","8","16"],["-1","-12","-48","-64"]],"row_sums_preview":["1","-5","25","-125","625","-3125","15625","-78125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A013611"],"edges":["e-95b12e1c49fd2aa5"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013611"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013611","chain":null}},{"id":"l-748bc5ed5dbb34c0","kind":"fingerprint_related","node_a_id":"n-1e5aa25413f55f87","node_b_id":"n-146f1cd501c36740","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-146f1cd501c36740","other":{"id":"n-146f1cd501c36740","name":"basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","8","16"],["1","12","48","64"],["1","16","96","256","256"]],"card_rows":[["1","8","16"],["1","12","48","64"],["1","16","96","256","256"],["1","20","160","640","1280","1024"]],"row_sums_preview":["25","125","625","3125","15625","78125","390625","1953125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013611"],"edges":["e-b91bf4c52fd06745"],"text":"basis-map coefficients of A112227 \u2192 basis A013611"},"headline":"basis-map coefficients of A112227 \u2192 basis A013611","chain":null}},{"id":"l-866e011ab9788155","kind":"fingerprint_related","node_a_id":"n-3dcaebc50e901d48","node_b_id":"n-1e5aa25413f55f87","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-3dcaebc50e901d48","other":{"id":"n-3dcaebc50e901d48","name":"Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","1"],["16","8","1"]],"card_rows":[["1"],["4","1"],["16","8","1"],["64","48","12","1"]],"row_sums_preview":["1","5","25","125","625","3125","15625","78125"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038231"],"derivation":{"root":"n-3dcaebc50e901d48","steps":[],"edges":[],"text":"A038231"},"headline":"A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","chain":null}},{"id":"l-879641c5e5f2b4ae","kind":"fingerprint_related","node_a_id":"n-1e5aa25413f55f87","node_b_id":"n-872d1ad8224c1da6","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-872d1ad8224c1da6","other":{"id":"n-872d1ad8224c1da6","name":"basis_map_A013611 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4"],["2","16","32"]],"card_rows":[["1"],["1","4"],["2","16","32"],["6","72","288","384"]],"row_sums_preview":["1","5","50","750","15000","375000","11250000","393750000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013611"],"edges":["e-d759043f254c4b9e"],"text":"A136572 \u2192 basis A013611"},"headline":"A136572 \u2192 basis A013611","chain":null}},{"id":"l-89bbf53d460c1e7e","kind":"fingerprint_related","node_a_id":"n-2e0409c0014ec240","node_b_id":"n-1e5aa25413f55f87","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-2e0409c0014ec240","other":{"id":"n-2e0409c0014ec240","name":"Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4"],["1","8","16"]],"card_rows":[["1"],["1","4"],["1","8","16"],["1","12","48","64"]],"row_sums_preview":["1","5","25","125","625","3125","15625","78125"],"regions":["(-inf, 0)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A013611"],"derivation":{"root":"n-2e0409c0014ec240","steps":[],"edges":[],"text":"A013611"},"headline":"A013611: Triangle of coefficients in expansion of (1+4x)^n.","chain":null}},{"id":"l-8aa34f2cbac16b22","kind":"fingerprint_related","node_a_id":"n-1e5aa25413f55f87","node_b_id":"n-7a0aecafcb4a073f","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-7a0aecafcb4a073f","other":{"id":"n-7a0aecafcb4a073f","name":"basis_map_A013611 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","first_index":1,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-4"],["1","8","16"]],"card_rows":[["1"],["-1","-4"],["1","8","16"],["-1","-12","-48","-64"]],"row_sums_preview":["1","-5","25","-125","625","-3125","15625","-78125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013611"],"edges":["e-f6ed0d45ba75837f"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013611"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013611","chain":null}},{"id":"l-8ac68ee03bded92f","kind":"fingerprint_related","node_a_id":"n-615df6ea54d3bbef","node_b_id":"n-1e5aa25413f55f87","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-615df6ea54d3bbef","other":{"id":"n-615df6ea54d3bbef","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","72","9"],["1728","1296","324","27"],["20736","20736","7776","1296","81"]],"card_rows":[["144","72","9"],["1728","1296","324","27"],["20736","20736","7776","1296","81"],["248832","311040","155520","38880","4860","243"]],"row_sums_preview":["225","3375","50625","759375","11390625","170859375","2562890625","38443359375"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":[],"edges":[],"text":"A038329"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","chain":null}},{"id":"l-afe1e5dd04912ba8","kind":"fingerprint_related","node_a_id":"n-8d4f9cf28f37366a","node_b_id":"n-1e5aa25413f55f87","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-8d4f9cf28f37366a","other":{"id":"n-8d4f9cf28f37366a","name":"derivative of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["8","32"]],"card_rows":[["4"],["8","32"],["12","96","192"],["16","192","768","1024"]],"row_sums_preview":["0","4","40","300","2000","12500","75000","437500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["D"],"edges":["e-b04b02754d5cebbf"],"text":"A013611 \u2192 D"},"headline":"A013611 \u2192 D","chain":null}},{"id":"l-bee3ca78e1bc47e8","kind":"fingerprint_related","node_a_id":"n-ee879560f84ab2b6","node_b_id":"n-1e5aa25413f55f87","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-ee879560f84ab2b6","other":{"id":"n-ee879560f84ab2b6","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","128","16"],["4096","3072","768","64"],["65536","65536","24576","4096","256"]],"card_rows":[["256","128","16"],["4096","3072","768","64"],["65536","65536","24576","4096","256"],["1048576","1310720","655360","163840","20480","1024"]],"row_sums_preview":["400","8000","160000","3200000","64000000","1280000000","25600000000","512000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A007318"],"edges":["e-7b68ca495e98f7ea"],"text":"A038330 \u2192 basis A007318"},"headline":"A038330 \u2192 basis A007318","chain":null}},{"id":"l-cf43087d7b10d383","kind":"fingerprint_related","node_a_id":"n-a4775661709ca859","node_b_id":"n-1e5aa25413f55f87","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-a4775661709ca859","other":{"id":"n-a4775661709ca859","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","2"],["64","32","4"]],"card_rows":[["1"],["8","2"],["64","32","4"],["512","384","96","8"]],"row_sums_preview":["1","10","100","1000","10000","100000","1000000","10000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013609"],"edges":["e-13f16c9d95bb60f6"],"text":"A027466 \u2192 basis A013609"},"headline":"A027466 \u2192 basis A013609","chain":null}},{"id":"l-d308875980a22ea3","kind":"fingerprint_related","node_a_id":"n-1e5aa25413f55f87","node_b_id":"n-c96fe4c1be387a19","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-c96fe4c1be387a19","other":{"id":"n-c96fe4c1be387a19","name":"basis_map_A013611 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","8"],["4","32","64"]],"card_rows":[["1"],["2","8"],["4","32","64"],["8","96","384","512"]],"row_sums_preview":["1","10","100","1000","10000","100000","1000000","10000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013611"],"edges":["e-d7d2422771947afc"],"text":"A130123 \u2192 basis A013611"},"headline":"A130123 \u2192 basis A013611","chain":null}},{"id":"l-d419aff7187c4cea","kind":"fingerprint_related","node_a_id":"n-1e5aa25413f55f87","node_b_id":"n-6e0864853f49fb26","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-6e0864853f49fb26","other":{"id":"n-6e0864853f49fb26","name":"basis_map_A013611 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-4"],["2","16","32"]],"card_rows":[["1"],["-1","-4"],["2","16","32"],["-6","-72","-288","-384"]],"row_sums_preview":["1","-5","50","-750","15000","-375000","11250000","-393750000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["basis A013611"],"edges":["e-e262ea1acc16711f"],"text":"basis-map coefficients of A021012 \u2192 basis A013611"},"headline":"basis-map coefficients of A021012 \u2192 basis A013611","chain":null}},{"id":"l-151c28aab89ce2a5","kind":"oeis_match","node_a_id":"n-1e5aa25413f55f87","node_b_id":null,"fingerprint":null,"oeis_id":"A074701","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":2,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A074701","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-63dae9e41b6e98fd","kind":"oeis_match","node_a_id":"n-1e5aa25413f55f87","node_b_id":null,"fingerprint":null,"oeis_id":"A038231","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A038231","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-9a72206c327da70c","kind":"oeis_match","node_a_id":"n-1e5aa25413f55f87","node_b_id":null,"fingerprint":null,"oeis_id":"A000351","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000351","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b7a729eab2ae468d","kind":"oeis_match","node_a_id":"n-1e5aa25413f55f87","node_b_id":null,"fingerprint":null,"oeis_id":"A384565","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A384565","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ca78ce57f2c8ed49","kind":"oeis_match","node_a_id":"n-1e5aa25413f55f87","node_b_id":null,"fingerprint":null,"oeis_id":"A013611","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A013611","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A038231","relation":"match_reversed_triangle"},{"oeis_id":"A000351","relation":"match_row_sums"},{"oeis_id":"A074701","relation":"match_row_sums"},{"oeis_id":"A384565","relation":"match_row_sums"},{"oeis_id":"A013611","relation":"match_triangle"}],"bfiles":[{"node_id":"n-1e5aa25413f55f87","oeis_id":"A000351","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":14900},{"node_id":"n-1e5aa25413f55f87","oeis_id":"A013611","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":19196},{"node_id":"n-1e5aa25413f55f87","oeis_id":"A038231","target":"reversed_triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":19196}],"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: 1,  4;\n   3: 1,  8,  16;\n   4: 1, 12,  48,   64;\n   5: 1, 16,  96,  256,   256;\n   6: 1, 20, 160,  640,  1280,   1024;\n   7: 1, 24, 240, 1280,  3840,   6144,   4096;\n   8: 1, 28, 336, 2240,  8960,  21504,  28672,  16384;\n   9: 1, 32, 448, 3584, 17920,  57344, 114688, 131072,  65536;\n  10: 1, 36, 576, 5376, 32256, 129024, 344064, 589824, 589824, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+4*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + 4*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; for(n = 2, N, my(i = n); v[i] = (1 + 4*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}},{"oeis_id":"A013611","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  4;\n  2: 1,  8,  16;\n  3: 1, 12,  48,   64;\n  4: 1, 16,  96,  256,   256;\n  5: 1, 20, 160,  640,  1280,   1024;\n  6: 1, 24, 240, 1280,  3840,   6144,   4096;\n  7: 1, 28, 336, 2240,  8960,  21504,  28672,  16384;\n  8: 1, 32, 448, 3584, 17920,  57344, 114688, 131072,  65536;\n  9: 1, 36, 576, 5376, 32256, 129024, 344064, 589824, 589824, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+4*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[(1 + 4*x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (1 + 4*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A038231","target":"reversed_triangle","first_row":1,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      4,      1;\n  2:     16,      8,      1;\n  3:     64,     48,     12,      1;\n  4:    256,    256,     96,     16,      1;\n  5:   1024,   1280,    640,    160,     20,     1;\n  6:   4096,   6144,   3840,   1280,    240,    24,    1;\n  7:  16384,  28672,  21504,   8960,   2240,   336,   28,   1;\n  8:  65536, 131072, 114688,  57344,  17920,  3584,  448,  32,  1;\n  9: 262144, 589824, 589824, 344064, 129024, 32256, 5376, 576, 36, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+4*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[(1 + 4*x)*P[n - 1, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (1 + 4*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013611"],"edges":["e-f93ad378f9476e41"],"text":"basis-map coefficients of A026729 \u2192 basis A013611"},"headline":"basis-map coefficients of A026729 \u2192 basis A013611","chain":null}