{"id":"n-0bb1f53c9b1ccec9","revision":0,"first_index":0,"fingerprint":"f6-49d880ced4b36b3a","name":"reversal of shift_x_plus_one of Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 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":true,"rows":[["1"],["1","3"],["1","6","8"],["1","9","24","20"],["1","12","48","80","48"],["1","15","80","200","240","112"],["1","18","120","400","720","672","256"],["1","21","168","700","1680","2352","1792","576"],["1","24","224","1120","3360","6272","7168","4608","1280"],["1","27","288","1680","6048","14112","21504","20736","11520","2816"],["1","30","360","2400","10080","28224","53760","69120","57600","28160","6144"],["1","33","440","3300","15840","51744","118272","190080","211200","154880","67584","13312"],["1","36","528","4400","23760","88704","236544","456192","633600","619520","405504","159744","28672"],["1","39","624","5720","34320","144144","439296","988416","1647360","2013440","1757184","1038336","372736","61440"],["1","42","728","7280","48048","224224","768768","1976832","3843840","5637632","6150144","4845568","2609152","860160","131072"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","15","54","189","648","2187","7290","24057","78732","255879","826686","2657205","8503056","27103491"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3","-2"],["0","0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174546,"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-b0ebae2b83172fc9"]},{"id":174547,"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-b0ebae2b83172fc9"]},{"id":174548,"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-b0ebae2b83172fc9"]}],"region_exact":{},"incoming_edges":[{"id":"e-b0ebae2b83172fc9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-0bb1f53c9b1ccec9","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"abd69df6ae2a7af1f289e0745acef01e20244caba6bd0899c13cb492261cb146","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-4b03a2849503b9e9","role":"sequence","name":"shift_x_plus_one of Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 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-d9899e204611ea31","kind":"candidate_equal","node_a_id":"n-f4f57d2aaca23cda","node_b_id":"n-0bb1f53c9b1ccec9","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-f4f57d2aaca23cda","other":{"id":"n-f4f57d2aaca23cda","name":"mobius_x_over_x_plus_one of Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","3"],["1","6","8"]],"card_rows":[["1"],["1","3"],["1","6","8"],["1","9","24","20"]],"row_sums_preview":["1","4","15","54","189","648","2187","7290"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-801895951aa20f29","steps":["x \u21a6 x/(x+1)"],"edges":["e-56e8a5a5f734c31a"],"text":"A103406 \u2192 x \u21a6 x/(x+1)"},"headline":"A103406 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-1225e6f466ad6c80","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-12985dc6a610374e","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-12985dc6a610374e","other":{"id":"n-12985dc6a610374e","name":"one_plus_derivative of basis_map_A007318 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":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","1"],["16","12","2"]],"card_rows":[["1"],["3","1"],["16","12","2"],["100","120","45","5"]],"row_sums_preview":["1","4","30","270","2646","27216","288684","3127410"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["x \u21a6 x+1","D+1"],"edges":["e-45645d17dad7cf5c","e-77a1767cd1de01a7"],"text":"A085880 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A085880 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}},{"id":"l-12d0f948cff92f44","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-bc9a904ef121dcaf","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-bc9a904ef121dcaf","other":{"id":"n-bc9a904ef121dcaf","name":"one_plus_derivative of basis_map_A007318 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","2"],["48","36","6"]],"card_rows":[["1"],["6","2"],["48","36","6"],["400","480","180","20"]],"row_sums_preview":["1","8","90","1080","13230","163296","2020788","25019280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["x \u21a6 x+1","D+1"],"edges":["e-0f8327ed3206c9f7","e-e23bc3194b56d102"],"text":"A356546 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A356546 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}},{"id":"l-1b4ae98be1445b31","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-3ae1126e4bd3090b","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-3ae1126e4bd3090b","other":{"id":"n-3ae1126e4bd3090b","name":"one_plus_derivative of basis_map_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","1"],["24","18","3"]],"card_rows":[["1"],["3","1"],["24","18","3"],["300","360","135","15"]],"row_sums_preview":["1","4","45","810","19845","612360","22733865","985134150"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A007318","D+1"],"edges":["e-5048ce3ebd740a6f","e-c74ef3b34cfcfdc5"],"text":"A085881 \u2192 basis A007318 \u2192 D+1"},"headline":"A085881 \u2192 basis A007318 \u2192 D+1","chain":null}},{"id":"l-1edc7bd3d22368f8","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-e83e0578ec753805","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-e83e0578ec753805","other":{"id":"n-e83e0578ec753805","name":"one_plus_derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["512","384","64"],["10240","12288","4608","512"],["196608","327680","196608","49152","4096"]],"card_rows":[["512","384","64"],["10240","12288","4608","512"],["196608","327680","196608","49152","4096"],["3670016","7864320","6553600","2621440","491520","32768"]],"row_sums_preview":["960","27648","774144","21233664","573308928","15288238080","403609485312","10567230160896"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013609","D+1"],"edges":["e-6efc8fa5932928ce","e-bce8eb484cc5237a"],"text":"A038330 \u2192 basis A013609 \u2192 D+1"},"headline":"A038330 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-288524a97c6f9448","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-ca753a595e484069","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-ca753a595e484069","other":{"id":"n-ca753a595e484069","name":"one_plus_derivative of basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","4"],["80","60","10"]],"card_rows":[["1"],["12","4"],["80","60","10"],["400","480","180","20"]],"row_sums_preview":["1","16","150","1080","6615","36288","183708","874800"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A007318","D+1"],"edges":["e-2fa6de2302a1c958","e-abeda2efcfd26e51"],"text":"A178820 \u2192 basis A007318 \u2192 D+1"},"headline":"A178820 \u2192 basis A007318 \u2192 D+1","chain":null}},{"id":"l-2b83441ddc7cefe4","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-f5d286d0e89e0b43","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-f5d286d0e89e0b43","other":{"id":"n-f5d286d0e89e0b43","name":"one_plus_derivative of basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","6"],["168","126","21"]],"card_rows":[["1"],["18","6"],["168","126","21"],["1120","1344","504","56"]],"row_sums_preview":["1","24","315","3024","23814","163296","1010394","5773680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["x \u21a6 x+1","D+1"],"edges":["e-a52727d7b51dd683","e-8d64603b189276f9"],"text":"A178822 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A178822 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}},{"id":"l-2e475bf89d10241c","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-2bc6e56cc1046162","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-2bc6e56cc1046162","other":{"id":"n-2bc6e56cc1046162","name":"one_plus_derivative of shift_x_plus_one of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["6","2"]],"card_rows":[["1"],["6","2"],["24","18","3"],["80","96","36","4"]],"row_sums_preview":["0","1","8","45","216","945","3888","15309"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["x \u21a6 x+1","D+1"],"edges":["e-1a916fd0c60fb95f","e-eb26f57388237012"],"text":"A003506 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A003506 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}},{"id":"l-4fe011ca3c50ee1e","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-9c056383ddb5a3c8","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-9c056383ddb5a3c8","other":{"id":"n-9c056383ddb5a3c8","name":"one_plus_derivative of derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["18","6"]],"card_rows":[["3"],["18","6"],["72","54","9"],["240","288","108","12"]],"row_sums_preview":["0","3","24","135","648","2835","11664","45927"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["D","D+1"],"edges":["e-d19cf3982d12ec2e","e-eff13a0c3dae11b6"],"text":"A120910 \u2192 D \u2192 D+1"},"headline":"A120910 \u2192 D \u2192 D+1","chain":null}},{"id":"l-651af3a70fe2b797","kind":"fingerprint_related","node_a_id":"n-878bf8921be6069e","node_b_id":"n-0bb1f53c9b1ccec9","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-878bf8921be6069e","other":{"id":"n-878bf8921be6069e","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","1"],["8","6","1"]],"card_rows":[["1"],["3","1"],["8","6","1"],["20","24","9","1"]],"row_sums_preview":["1","4","15","54","189","648","2187","7290"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-af754f07daf65b5e","steps":["D+1"],"edges":["e-ecf027aa2d377831"],"text":"A038207 \u2192 D+1"},"headline":"A038207 \u2192 D+1","chain":null}},{"id":"l-6aa821e3226e9cc9","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-952f6c3d4cddeec0","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-952f6c3d4cddeec0","other":{"id":"n-952f6c3d4cddeec0","name":"one_plus_derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","3"],["72","54","9"]],"card_rows":[["1"],["9","3"],["72","54","9"],["540","648","243","27"]],"row_sums_preview":["1","12","135","1458","15309","157464","1594323","15943230"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["x \u21a6 x+1","D+1"],"edges":["e-02e8db4659f506e0","e-92d11df847ff8185"],"text":"A038221 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A038221 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}},{"id":"l-7733c43328e940bb","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-1cc2169a82f1af75","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-1cc2169a82f1af75","other":{"id":"n-1cc2169a82f1af75","name":"one_plus_derivative of basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","5"],["120","90","15"]],"card_rows":[["1"],["15","5"],["120","90","15"],["700","840","315","35"]],"row_sums_preview":["1","20","225","1890","13230","81648","459270","2405700"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A007318","D+1"],"edges":["e-484d449f4d0f99c5","e-ebb1094e4fe5f947"],"text":"A178821 \u2192 basis A007318 \u2192 D+1"},"headline":"A178821 \u2192 basis A007318 \u2192 D+1","chain":null}},{"id":"l-8c2e38028b074f17","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-381980da7221376f","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-381980da7221376f","other":{"id":"n-381980da7221376f","name":"one_plus_derivative of shift_x_plus_one of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","4"],["72","54","9"]],"card_rows":[["1"],["12","4"],["72","54","9"],["320","384","144","16"]],"row_sums_preview":["1","16","135","864","4725","23328","107163","466560"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-375ba152c4aebf61","steps":["x \u21a6 x+1","D+1"],"edges":["e-18ace392e620d7bd","e-98cf6c45bc3abde0"],"text":"A135065 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A135065 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}},{"id":"l-8e9f35063c7a55b7","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-6bd0da8cabe63b8e","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-6bd0da8cabe63b8e","other":{"id":"n-6bd0da8cabe63b8e","name":"one_plus_derivative of basis_map_A007318 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","2"],["48","36","6"]],"card_rows":[["1"],["6","2"],["48","36","6"],["480","576","216","24"]],"row_sums_preview":["1","8","90","1296","22680","466560","11022480","293932800"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A007318","D+1"],"edges":["e-bd56c468ffc76132","e-3d66d9d55295c71a"],"text":"A156992 \u2192 basis A007318 \u2192 D+1"},"headline":"A156992 \u2192 basis A007318 \u2192 D+1","chain":null}},{"id":"l-9bc2da8b679319a9","kind":"fingerprint_related","node_a_id":"n-4b03a2849503b9e9","node_b_id":"n-0bb1f53c9b1ccec9","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-4b03a2849503b9e9","other":{"id":"n-4b03a2849503b9e9","name":"shift_x_plus_one of Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","1"],["8","6","1"]],"card_rows":[["1"],["3","1"],["8","6","1"],["20","24","9","1"]],"row_sums_preview":["1","4","15","54","189","648","2187","7290"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f24dc5f77f658331","steps":["x \u21a6 x+1"],"edges":["e-09eb0f66de37936d"],"text":"A093375 \u2192 x \u21a6 x+1"},"headline":"A093375 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-a9819717141b61ae","kind":"fingerprint_related","node_a_id":"n-e924c6dc85f754f0","node_b_id":"n-0bb1f53c9b1ccec9","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-e924c6dc85f754f0","other":{"id":"n-e924c6dc85f754f0","name":"shift_x_plus_one of one_plus_derivative of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["6","2"]],"card_rows":[["1"],["6","2"],["24","18","3"],["80","96","36","4"]],"row_sums_preview":["0","1","8","45","216","945","3888","15309"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f24dc5f77f658331","steps":["D","basis A007318"],"edges":["e-2695bc03a925fb68","e-35166d38e88e9907"],"text":"A093375 \u2192 D \u2192 basis A007318"},"headline":"A093375 \u2192 D \u2192 basis A007318","chain":null}},{"id":"l-b16e6c75f1b85b92","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-c11a77a101be52d0","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-c11a77a101be52d0","other":{"id":"n-c11a77a101be52d0","name":"one_plus_derivative of shift_x_plus_one of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","3"],["48","36","6"]],"card_rows":[["1"],["9","3"],["48","36","6"],["200","240","90","10"]],"row_sums_preview":["1","12","90","540","2835","13608","61236","262440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f1b09730105c8f0","steps":["x \u21a6 x+1","D+1"],"edges":["e-34e3d9a08d46fb54","e-6bdd1daa8919005b"],"text":"A094305 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A094305 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}},{"id":"l-b77e07ad93bb57a8","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-0d46ef1f3325daef","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-0d46ef1f3325daef","other":{"id":"n-0d46ef1f3325daef","name":"one_plus_derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1152","864","144"],["34560","41472","15552","1728"],["995328","1658880","995328","248832","20736"]],"card_rows":[["1152","864","144"],["34560","41472","15552","1728"],["995328","1658880","995328","248832","20736"],["27869184","59719680","49766400","19906560","3732480","248832"]],"row_sums_preview":["2160","93312","3919104","161243136","6530347008","261213880320","10344069660672","406239826673664"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["x \u21a6 x+1","D+1"],"edges":["e-240272ddc3b4d83d","e-774a7e7a58b8f49c"],"text":"A038338 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A038338 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}},{"id":"l-c22b78fd7ee451f5","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-07eaafd57b072940","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-07eaafd57b072940","other":{"id":"n-07eaafd57b072940","name":"one_plus_derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","2"],["32","24","4"]],"card_rows":[["1"],["6","2"],["32","24","4"],["160","192","72","8"]],"row_sums_preview":["1","8","60","432","3024","20736","139968","933120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A007318","D+1"],"edges":["e-1756f066a00732e3","e-4444b187152e1076"],"text":"A038208 \u2192 basis A007318 \u2192 D+1"},"headline":"A038208 \u2192 basis A007318 \u2192 D+1","chain":null}},{"id":"l-c8fcb7c5ebe04906","kind":"fingerprint_related","node_a_id":"n-8cd21ebd6c5c32f0","node_b_id":"n-0bb1f53c9b1ccec9","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-8cd21ebd6c5c32f0","other":{"id":"n-8cd21ebd6c5c32f0","name":"one_plus_derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["9","3"],["24","18","3"]],"card_rows":[["3"],["9","3"],["24","18","3"],["60","72","27","3"]],"row_sums_preview":["3","12","45","162","567","1944","6561","21870"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["D+1"],"edges":["e-879e9fec43b0fba4"],"text":"A120910 \u2192 D+1"},"headline":"A120910 \u2192 D+1","chain":null}},{"id":"l-d434ea7f99a49ada","kind":"fingerprint_related","node_a_id":"n-de5e14d043ee5c31","node_b_id":"n-0bb1f53c9b1ccec9","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-de5e14d043ee5c31","other":{"id":"n-de5e14d043ee5c31","name":"one_plus_derivative of shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-3","-1"],["8","6","1"]],"card_rows":[["1"],["-3","-1"],["8","6","1"],["-20","-24","-9","-1"]],"row_sums_preview":["1","-4","15","-54","189","-648","2187","-7290"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["x \u21a6 x+1","D+1"],"edges":["e-51a320c1f4e00092","e-98d98e7ca93f8414"],"text":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}},{"id":"l-d695cb9d222c0fb6","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-14414e2c5a7f4554","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-14414e2c5a7f4554","other":{"id":"n-14414e2c5a7f4554","name":"one_plus_derivative of basis_map_A007318 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","1"],["16","12","2"]],"card_rows":[["1"],["3","1"],["16","12","2"],["120","144","54","6"]],"row_sums_preview":["1","4","30","324","4536","77760","1574640","36741600"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["x \u21a6 x+1","D+1"],"edges":["e-0b89a3d3e0cfd72c","e-9f3b51c7c6411820"],"text":"A196347 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A196347 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}},{"id":"l-e85c7d98e12e7035","kind":"fingerprint_related","node_a_id":"n-080a737a003c22ab","node_b_id":"n-0bb1f53c9b1ccec9","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-080a737a003c22ab","other":{"id":"n-080a737a003c22ab","name":"one_plus_derivative of shift_x_plus_one of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["6","2"],["16","12","2"]],"card_rows":[["2"],["6","2"],["16","12","2"],["40","48","18","2"]],"row_sums_preview":["2","8","30","108","378","1296","4374","14580"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A007318","D+1"],"edges":["e-014dd8a0f9594ad5","e-3ed314a329a43a00"],"text":"A028326 \u2192 basis A007318 \u2192 D+1"},"headline":"A028326 \u2192 basis A007318 \u2192 D+1","chain":null}},{"id":"l-ffc90b3365dc582f","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-7a7453614630a4b8","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-7a7453614630a4b8","other":{"id":"n-7a7453614630a4b8","name":"one_plus_derivative of basis_map_A007318 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["33","11"],["968","726","121"]],"card_rows":[["1"],["33","11"],["968","726","121"],["26620","31944","11979","1331"]],"row_sums_preview":["1","44","1815","71874","2767149","104361048","3874403907","142061476590"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A007318","D+1"],"edges":["e-a5d0659f5315d3b7","e-61943e16b146f4ab"],"text":"A038325 \u2192 basis A007318 \u2192 D+1"},"headline":"A038325 \u2192 basis A007318 \u2192 D+1","chain":null}},{"id":"l-fff3ce5b766758f4","kind":"fingerprint_related","node_a_id":"n-0bb1f53c9b1ccec9","node_b_id":"n-eb1c168b96e7421e","fingerprint":"f6-49d880ced4b36b3a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49d880ced4b36b3a"},"other_id":"n-eb1c168b96e7421e","other":{"id":"n-eb1c168b96e7421e","name":"one_plus_derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-49d880ced4b36b3a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","4"],["128","96","16"]],"card_rows":[["1"],["12","4"],["128","96","16"],["1280","1536","576","64"]],"row_sums_preview":["1","16","240","3456","48384","663552","8957952","119439360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013611","D+1"],"edges":["e-267942f18014ef6e","e-d119f325b906f14a"],"text":"A027466 \u2192 basis A013611 \u2192 D+1"},"headline":"A027466 \u2192 basis A013611 \u2192 D+1","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: 1,  3;\n  2: 1,  6,   8;\n  3: 1,  9,  24,   20;\n  4: 1, 12,  48,   80,   48;\n  5: 1, 15,  80,  200,  240,   112;\n  6: 1, 18, 120,  400,  720,   672,   256;\n  7: 1, 21, 168,  700, 1680,  2352,  1792,   576;\n  8: 1, 24, 224, 1120, 3360,  6272,  7168,  4608,  1280;\n  9: 1, 27, 288, 1680, 6048, 14112, 21504, 20736, 11520, 2816;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+3*x+(2*n-2)*x^2)*P(n-1,x) - (x^2+2*x^3)*d/dx 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 + 3*x + (2*n - 2)*x^2)*P[n - 1, x] - (x^2 + 2*x^3)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {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 + 3*x + (2*n - 2)*x^2)*v[i - 1] - (x^2 + 2*x^3)*deriv(v[i - 1], 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-f24dc5f77f658331","steps":["x \u21a6 x+1","reverse"],"edges":["e-09eb0f66de37936d","e-b0ebae2b83172fc9"],"text":"A093375 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A093375 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}
