{"id":"n-146f1cd501c36740","revision":1,"first_index":1,"fingerprint":"f6-03d7b8417365c509","name":"basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.","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","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"],["1","60","1680","29120","349440","3075072","20500480","105431040","421724160","1312030720","3148873728","5725224960","7633633280","7046430720","4026531840","1073741824"],["1","64","1920","35840","465920","4472832","32800768","187432960","843448320","2998927360","8396996608","18320719872","30534533120","37580963840","32212254720","17179869184","4294967296"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["25","125","625","3125","15625","78125","390625","1953125","9765625","48828125","244140625","1220703125","6103515625","30517578125","152587890625"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1","8","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167618,"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-b91bf4c52fd06745"]},{"id":167619,"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-b91bf4c52fd06745"]},{"id":167620,"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-b91bf4c52fd06745"]}],"region_exact":{},"incoming_edges":[{"id":"e-b91bf4c52fd06745","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-146f1cd501c36740","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":"0dbc7353cbaae390847a6d29468844e253dc4966f8f5a8500bb862b8dbbeb295","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-5fe240122e98ffb3","role":"sequence","name":"basis-map coefficients of A scaled Hermite triangle.","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-0fb875f722d671b7","kind":"fingerprint_related","node_a_id":"n-146f1cd501c36740","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-12bc9123d9dd2616","kind":"fingerprint_related","node_a_id":"n-8547d44959e7b9a0","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-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-2306c220625972f3","kind":"fingerprint_related","node_a_id":"n-146f1cd501c36740","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-26d74625329b923f","kind":"fingerprint_related","node_a_id":"n-8d4f9cf28f37366a","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-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-40554df21a820750","kind":"fingerprint_related","node_a_id":"n-ef5c9478e7d83dab","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-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-596e426b434a7b9c","kind":"fingerprint_related","node_a_id":"n-a4775661709ca859","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-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-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-1e5aa25413f55f87","other":{"id":"n-1e5aa25413f55f87","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.","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"],["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-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}},{"id":"l-8e3afb35f87a52ff","kind":"fingerprint_related","node_a_id":"n-ee879560f84ab2b6","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-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-b8a4ae4524ba840b","kind":"fingerprint_related","node_a_id":"n-7a0aecafcb4a073f","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-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-bb758825bb43bfd0","kind":"fingerprint_related","node_a_id":"n-615df6ea54d3bbef","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-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-ce8bbaa2f67e8593","kind":"fingerprint_related","node_a_id":"n-146f1cd501c36740","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-e0b8d6a7c249f916","kind":"fingerprint_related","node_a_id":"n-3dcaebc50e901d48","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-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-e7d4544a1c8c0322","kind":"fingerprint_related","node_a_id":"n-146f1cd501c36740","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-f0adfdfab939401d","kind":"fingerprint_related","node_a_id":"n-2e0409c0014ec240","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-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-78c164a4ece369a5","kind":"oeis_match","node_a_id":"n-146f1cd501c36740","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000351","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f5005ace99cb6c35","kind":"oeis_match","node_a_id":"n-146f1cd501c36740","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":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A074701","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A000351","relation":"match_row_sums"},{"oeis_id":"A074701","relation":"match_row_sums"}],"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,  8,  16;\n   2: 1, 12,  48,    64;\n   3: 1, 16,  96,   256,   256;\n   4: 1, 20, 160,   640,  1280,   1024;\n   5: 1, 24, 240,  1280,  3840,   6144,    4096;\n   6: 1, 28, 336,  2240,  8960,  21504,   28672,   16384;\n   7: 1, 32, 448,  3584, 17920,  57344,  114688,  131072,    65536;\n   8: 1, 36, 576,  5376, 32256, 129024,  344064,  589824,   589824,   262144;\n   9: 1, 40, 720,  7680, 53760, 258048,  860160, 1966080,  2949120,  2621440,  1048576;\n  10: 1, 44, 880, 10560, 84480, 473088, 1892352, 5406720, 10813440, 14417920, 11534336, 4194304;\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 + 8*x + 16*x^2.","mathematica":"P[1, x_] := 1 + 8*x + 16*x^2;\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 + 8*x + 16*x^2; 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}}],"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}