{"id":"n-78a31084f0075a3e","revision":0,"first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","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":[["1760","3872"],["52800","232320","255552"],["1408000","9292800","20444160","14992384"],["35200000","309760000","1022208000","1499238400","824581120"],["844800000","9292800000","40888320000","89954304000","98949734400","43537883136"],["19712000000","260198400000","1431091200000","4197867520000","6926481408000","6095303639040","2234944667648"],["450560000000","6938624000000","45794918400000","167914700800000","369412341760000","487624291123200","357591146823680","112385789001728"],["10137600000000","178421760000000","1373847552000000","6044929228800000","16623555379200000","29257457467392000","32183203214131200","20229442020311040","5563096555585536"],["225280000000000","4460544000000000","39252787200000000","201497640960000000","664942215168000000","1462872873369600000","2145546880942080000","2022944202031104000","1112619311117107200","271973609384181760"],["4956160000000000","109035520000000000","1079451648000000000","6332783001600000000","24381214556160000000","64366406428262400000","118005078451814400000","148349241482280960000","122388124222881792000","59834194064519987200","13163522694194397184"],["108134400000000000","2616852480000000000","28785377280000000000","189983490048000000000","835927356211200000000","2574656257130496000000","5664243765687091200000","8900954488936857600000","9791049937830543360000","7180103287742398464000","3159245446606655324160","631849089321331064832"],["2342912000000000000","61852876800000000000","748419809280000000000","5488411934720000000000","27167639076864000000000","95630089550561280000000","245450563179773952000000","462849633424716595200000","636418245958985318400000","622275618271007866880000","410701908058865192140800","164280763223546076856320","30118139924316780756992"],["50462720000000000000","1443233792000000000000","19050686054400000000000","153675534172160000000000","845215437946880000000000","3347053134269644800000000","9818022527190958080000000","21599649559820107776000000","35639421773703177830400000","43559293278970550681600000","38332178085494084599808000","22999306851296450759884800","8433079178808698611957760","1427136476413779765100544"],["1081344000000000000000","33305395200000000000000","476267151360000000000000","4191150931968000000000000","25356463138406400000000000","111568437808988160000000000","368175844769660928000000000","925699266849433190400000000","1781971088685158891520000000","2613557596738233040896000000","2874913356412056344985600000","2299930685129645075988480000","1264961876821304791793664000","428140942924133929530163200","67279291030935331783311360"],["23068672000000000000000","761266176000000000000000","11723499110400000000000000","111764024852480000000000000","737642564026368000000000000","3570190009887621120000000000","13090696702921277440000000000","37027970673977327616000000000","81461535482750120755200000000","139389738492705762181120000000","183994454810371606079078400000","183994454810371606079078400000","134929266860939177791324160000","68502550867861428724826112000","21529373129899306170659635200","3157641392385231571696746496"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["5632","540672","46137344","3690987520","283467841536","21165598834688","1548112371908608","111464090777419776","7926335344172072960","558014008229713936384","38959523483674573012992","2701193628201437062234112","186174576220652892904751104","12766256655130484084897218560","871509787656907713528983453696"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1760","3872"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["40","88"],["20","44"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195148,"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-7a4ada96399020f0"]},{"id":195149,"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-7a4ada96399020f0"]},{"id":195150,"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-7a4ada96399020f0"]},{"id":195151,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-7a4ada96399020f0"]}],"region_exact":{},"incoming_edges":[{"id":"e-7a4ada96399020f0","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-78a31084f0075a3e","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"8c7856d8ae0686f35d8039fb5f6ab49d2996ff8aa220ac41edf165044ed50dee","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-586959989bac3453","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","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-23578e925d5135c1","kind":"fingerprint_related","node_a_id":"n-a11e46881092378a","node_b_id":"n-78a31084f0075a3e","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-a11e46881092378a","other":{"id":"n-a11e46881092378a","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["484","440","100"],["10648","14520","6600","1000"],["234256","425920","290400","88000","10000"]],"card_rows":[["484","440","100"],["10648","14520","6600","1000"],["234256","425920","290400","88000","10000"],["5153632","11712800","10648000","4840000","1100000","100000"]],"row_sums_preview":["1024","32768","1048576","33554432","1073741824","34359738368","1099511627776","35184372088832"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":["x \u21a6 x+1"],"edges":["e-0c3feb8686271cde"],"text":"A038336 \u2192 x \u21a6 x+1"},"headline":"A038336 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-55db7dc32334d78a","kind":"fingerprint_related","node_a_id":"n-5d1c1a2161cc3127","node_b_id":"n-78a31084f0075a3e","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-5d1c1a2161cc3127","other":{"id":"n-5d1c1a2161cc3127","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["440","200"],["14520","13200","3000"],["425920","580800","264000","40000"]],"card_rows":[["440","200"],["14520","13200","3000"],["425920","580800","264000","40000"],["11712800","21296000","14520000","4400000","500000"]],"row_sums_preview":["640","30720","1310720","52428800","2013265920","75161927680","2748779069440","98956046499840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":["x \u21a6 x+1","D"],"edges":["e-0c3feb8686271cde","e-1aff57056f3d2257"],"text":"A038336 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A038336 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-5caf972c07cd6cd4","kind":"fingerprint_related","node_a_id":"n-78a31084f0075a3e","node_b_id":"n-1b95b22bf8059e31","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-1b95b22bf8059e31","other":{"id":"n-1b95b22bf8059e31","name":"reversal of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1936","1760","400"],["85184","116160","52800","8000"],["3748096","6814720","4646400","1408000","160000"]],"card_rows":[["1936","1760","400"],["85184","116160","52800","8000"],["3748096","6814720","4646400","1408000","160000"],["164916224","374809600","340736000","154880000","35200000","3200000"]],"row_sums_preview":["4096","262144","16777216","1073741824","68719476736","4398046511104","281474976710656","18014398509481984"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cfc5b48d32898a3e","steps":["basis A013611","reverse"],"edges":["e-9ac6ce8d07a161c9","e-7e77526ede378673"],"text":"A038301 \u2192 basis A013611 \u2192 reverse"},"headline":"A038301 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-629d89b23a87315a","kind":"fingerprint_related","node_a_id":"n-65d5efd76862520a","node_b_id":"n-78a31084f0075a3e","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-65d5efd76862520a","other":{"id":"n-65d5efd76862520a","name":"shift_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","10"],["484","440","100"]],"card_rows":[["1"],["22","10"],["484","440","100"],["10648","14520","6600","1000"]],"row_sums_preview":["1","32","1024","32768","1048576","33554432","1073741824","34359738368"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-e169cb33ba7b2c8c","e-a1ded84a3ad1c29f"],"text":"A038271 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A038271 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-6fbc56dda1bc9aed","kind":"fingerprint_related","node_a_id":"n-4c3f91459b284843","node_b_id":"n-78a31084f0075a3e","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-4c3f91459b284843","other":{"id":"n-4c3f91459b284843","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["110","50"],["1815","1650","375"],["26620","36300","16500","2500"]],"card_rows":[["110","50"],["1815","1650","375"],["26620","36300","16500","2500"],["366025","665500","453750","137500","15625"]],"row_sums_preview":["160","3840","81920","1638400","31457280","587202560","10737418240","193273528320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["D"],"edges":["e-c0fdb556b5da1838"],"text":"A038319 \u2192 D"},"headline":"A038319 \u2192 D","chain":null}},{"id":"l-73f7b9980554b02f","kind":"fingerprint_related","node_a_id":"n-d46be0acd5f53980","node_b_id":"n-78a31084f0075a3e","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-d46be0acd5f53980","other":{"id":"n-d46be0acd5f53980","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["25","110","121"],["125","825","1815","1331"],["625","5500","18150","26620","14641"]],"card_rows":[["25","110","121"],["125","825","1815","1331"],["625","5500","18150","26620","14641"],["3125","34375","151250","332750","366025","161051"]],"row_sums_preview":["256","4096","65536","1048576","16777216","268435456","4294967296","68719476736"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["reverse"],"edges":["e-ad80611cbcc0e8c7"],"text":"A038319 \u2192 reverse"},"headline":"A038319 \u2192 reverse","chain":null}},{"id":"l-7bceaf45d2a70fc7","kind":"fingerprint_related","node_a_id":"n-586959989bac3453","node_b_id":"n-78a31084f0075a3e","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-586959989bac3453","other":{"id":"n-586959989bac3453","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","1760","1936"],["8000","52800","116160","85184"],["160000","1408000","4646400","6814720","3748096"]],"card_rows":[["400","1760","1936"],["8000","52800","116160","85184"],["160000","1408000","4646400","6814720","3748096"],["3200000","35200000","154880000","340736000","374809600","164916224"]],"row_sums_preview":["4096","262144","16777216","1073741824","68719476736","4398046511104","281474976710656","18014398509481984"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cfc5b48d32898a3e","steps":["basis A013611"],"edges":["e-9ac6ce8d07a161c9"],"text":"A038301 \u2192 basis A013611"},"headline":"A038301 \u2192 basis A013611","chain":null}},{"id":"l-8090c91b0a6accc8","kind":"fingerprint_related","node_a_id":"n-040ab732c43295a8","node_b_id":"n-78a31084f0075a3e","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-040ab732c43295a8","other":{"id":"n-040ab732c43295a8","name":"reversal of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","11"],["25","110","121"]],"card_rows":[["1"],["5","11"],["25","110","121"],["125","825","1815","1331"]],"row_sums_preview":["1","16","256","4096","65536","1048576","16777216","268435456"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e6ebe74518d42cbe","steps":["basis A007318","reverse"],"edges":["e-1194f7dc6f45a71b","e-366811f71221c5cf"],"text":"A038259 \u2192 basis A007318 \u2192 reverse"},"headline":"A038259 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-8cc67f9c302898ae","kind":"fingerprint_related","node_a_id":"n-78a31084f0075a3e","node_b_id":"n-436c06bbad4746ea","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-436c06bbad4746ea","other":{"id":"n-436c06bbad4746ea","name":"basis_map_A013610 of basis_map_A013610 of Triangle of coefficients in expansion of (1+11x)^n.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2025","8910","9801"],["91125","601425","1323135","970299"],["4100625","36085500","119082150","174653820","96059601"]],"card_rows":[["2025","8910","9801"],["91125","601425","1323135","970299"],["4100625","36085500","119082150","174653820","96059601"],["184528125","2029809375","8931161250","19648554750","21613410225","9509900499"]],"row_sums_preview":["20736","2985984","429981696","61917364224","8916100448256","1283918464548864","184884258895036416","26623333280885243904"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b1e393c17b0898c2","steps":["basis A013610","basis A013610"],"edges":["e-4217afbeac9f09d9","e-0ca3a617dc7dd8e3"],"text":"A013618 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A013618 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-91362bd633de595e","kind":"fingerprint_related","node_a_id":"n-2bc1923905556bf6","node_b_id":"n-78a31084f0075a3e","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-2bc1923905556bf6","other":{"id":"n-2bc1923905556bf6","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["121","110","25"],["1331","1815","825","125"],["14641","26620","18150","5500","625"]],"card_rows":[["121","110","25"],["1331","1815","825","125"],["14641","26620","18150","5500","625"],["161051","366025","332750","151250","34375","3125"]],"row_sums_preview":["256","4096","65536","1048576","16777216","268435456","4294967296","68719476736"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":[],"edges":[],"text":"A038319"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","chain":null}},{"id":"l-9c50d97177100ee7","kind":"fingerprint_related","node_a_id":"n-a85b59eef3df50f5","node_b_id":"n-78a31084f0075a3e","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-a85b59eef3df50f5","other":{"id":"n-a85b59eef3df50f5","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["100","440","484"],["1000","6600","14520","10648"],["10000","88000","290400","425920","234256"]],"card_rows":[["100","440","484"],["1000","6600","14520","10648"],["10000","88000","290400","425920","234256"],["100000","1100000","4840000","10648000","11712800","5153632"]],"row_sums_preview":["1024","32768","1048576","33554432","1073741824","34359738368","1099511627776","35184372088832"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-29d7c5a1e19b9d74","steps":["x \u21a6 x/(x+1)"],"edges":["e-40c9926c83cb24ec"],"text":"A038314 \u2192 x \u21a6 x/(x+1)"},"headline":"A038314 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-a7cc4b4a0599436e","kind":"fingerprint_related","node_a_id":"n-132cb39dd7c496a1","node_b_id":"n-78a31084f0075a3e","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-132cb39dd7c496a1","other":{"id":"n-132cb39dd7c496a1","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["5"],["110","50"]],"card_rows":[["5"],["110","50"],["1815","1650","375"],["26620","36300","16500","2500"]],"row_sums_preview":["0","5","160","3840","81920","1638400","31457280","587202560"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e6ebe74518d42cbe","steps":["basis A007318","D"],"edges":["e-1194f7dc6f45a71b","e-8535c3cecaab2b27"],"text":"A038259 \u2192 basis A007318 \u2192 D"},"headline":"A038259 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-d6c1446adec650ed","kind":"fingerprint_related","node_a_id":"n-803e5a0eeca7a5f4","node_b_id":"n-78a31084f0075a3e","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-803e5a0eeca7a5f4","other":{"id":"n-803e5a0eeca7a5f4","name":"reversal of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["100","440","484"],["1000","6600","14520","10648"],["10000","88000","290400","425920","234256"]],"card_rows":[["100","440","484"],["1000","6600","14520","10648"],["10000","88000","290400","425920","234256"],["100000","1100000","4840000","10648000","11712800","5153632"]],"row_sums_preview":["1024","32768","1048576","33554432","1073741824","34359738368","1099511627776","35184372088832"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":["x \u21a6 x+1","reverse"],"edges":["e-0c3feb8686271cde","e-e6d4a4dba1844cb2"],"text":"A038336 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A038336 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-d7690950f586523a","kind":"fingerprint_related","node_a_id":"n-44fec6014b2adb13","node_b_id":"n-78a31084f0075a3e","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-44fec6014b2adb13","other":{"id":"n-44fec6014b2adb13","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","11"],["25","110","121"]],"card_rows":[["1"],["5","11"],["25","110","121"],["125","825","1815","1331"]],"row_sums_preview":["1","16","256","4096","65536","1048576","16777216","268435456"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-32b2129ae50cd22c","e-26cb73d403c7e3fa"],"text":"A038220 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A038220 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-e66641bad117aabc","kind":"fingerprint_related","node_a_id":"n-0195873a37fb074a","node_b_id":"n-78a31084f0075a3e","fingerprint":"f6-2235b7dd30e0ad6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2235b7dd30e0ad6a"},"other_id":"n-0195873a37fb074a","other":{"id":"n-0195873a37fb074a","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","first_index":0,"fingerprint":"f6-2235b7dd30e0ad6a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","5"],["121","110","25"]],"card_rows":[["1"],["11","5"],["121","110","25"],["1331","1815","825","125"]],"row_sums_preview":["1","16","256","4096","65536","1048576","16777216","268435456"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e6ebe74518d42cbe","steps":["basis A007318"],"edges":["e-1194f7dc6f45a71b"],"text":"A038259 \u2192 basis A007318"},"headline":"A038259 \u2192 basis A007318","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:             1760,               3872;\n  1:            52800,             232320,              255552;\n  2:          1408000,            9292800,            20444160,            14992384;\n  3:         35200000,          309760000,          1022208000,          1499238400,            824581120;\n  4:        844800000,         9292800000,         40888320000,         89954304000,          98949734400,          43537883136;\n  5:      19712000000,       260198400000,       1431091200000,       4197867520000,        6926481408000,        6095303639040,         2234944667648;\n  6:     450560000000,      6938624000000,      45794918400000,     167914700800000,      369412341760000,      487624291123200,       357591146823680,       112385789001728;\n  7:   10137600000000,    178421760000000,    1373847552000000,    6044929228800000,    16623555379200000,    29257457467392000,     32183203214131200,     20229442020311040,      5563096555585536;\n  8:  225280000000000,   4460544000000000,   39252787200000000,  201497640960000000,   664942215168000000,  1462872873369600000,   2145546880942080000,   2022944202031104000,   1112619311117107200,   271973609384181760;\n  9: 4956160000000000, 109035520000000000, 1079451648000000000, 6332783001600000000, 24381214556160000000, 64366406428262400000, 118005078451814400000, 148349241482280960000, 122388124222881792000, 59834194064519987200, 13163522694194397184;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (20*n+40+(44*n+88)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1760 + 3872*x.","mathematica":"P[0, x_] := 1760 + 3872*x;\nP[n_, x_] := P[n, x] = Expand[Together[((20*n + 40 + (44*n + 88)*x)*P[n - 1, x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1760 + 3872*x; for(n = 1, N, my(i = n + 1); v[i] = ((20*n + 40 + (44*n + 88)*x)*v[i - 1])/(n + 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cfc5b48d32898a3e","steps":["basis A013611","D"],"edges":["e-9ac6ce8d07a161c9","e-7a4ada96399020f0"],"text":"A038301 \u2192 basis A013611 \u2192 D"},"headline":"A038301 \u2192 basis A013611 \u2192 D","chain":null}
