{"id":"n-1f922d5ae3a4fc22","revision":1,"first_index":0,"fingerprint":"f6-937db0732045a112","name":"veronese_section_2_1 of Riordan array (1-x,x/(1-x)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["0","1"],["0","2"],["0","3","1"],["0","4","4"],["0","5","10","1"],["0","6","20","6"],["0","7","35","21","1"],["0","8","56","56","8"],["0","9","84","126","36","1"],["0","10","120","252","120","10"],["0","11","165","462","330","55","1"],["0","12","220","792","792","220","12"],["0","13","286","1287","1716","715","78","1"],["0","14","364","2002","3432","2002","364","14"]],"degrees":[-1,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["0","1","2","4","8","16","32","64","128","256","512","1024","2048","4096","8192"],"nonn":true,"definitions":[{"position":0,"source":"derived:veronese_section_2_1","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163043,"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-6fdf1329e5e61ed8"]},{"id":163044,"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-6fdf1329e5e61ed8"]},{"id":163045,"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-6fdf1329e5e61ed8"]}],"region_exact":{},"incoming_edges":[{"id":"e-6fdf1329e5e61ed8","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-1f922d5ae3a4fc22","premise_count":1,"theorem":"Veronese sections of Polya frequency sequences (Aissen--Schoenberg--Whitney--Edrei): if P is real-rooted with coefficients of one sign, equivalently nonzero with all roots in (-inf, 0], every section V_(r,j)(P) is zero or real-rooted with roots in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"ce1e96040c94cbbf6f3cf18b11b7b9e392a94b5951f8849fc59a9f4da9c89f7a","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-7717563017cb9e5f","role":"sequence","name":"Riordan array (1-x,x/(1-x)).","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-22e54997adbec1ad","kind":"fingerprint_related","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":"n-01470b248c23fd02","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-01470b248c23fd02","other":{"id":"n-01470b248c23fd02","name":"veronese_section_2_1 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["5"],["30"]],"card_rows":[["5"],["30"],["105","35"],["280","280"]],"row_sums_preview":["0","5","30","140","560","2016","6720","21120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["Veronese 2 (j=1)"],"edges":["e-1471988cf5f0d7aa"],"text":"A178821 \u2192 Veronese 2 (j=1)"},"headline":"A178821 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-25128a115af5bdb3","kind":"fingerprint_related","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":"n-f6e8ebcef0e02821","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-f6e8ebcef0e02821","other":{"id":"n-f6e8ebcef0e02821","name":"veronese_section_2_1 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["4"],["20"]],"card_rows":[["4"],["20"],["60","20"],["140","140"]],"row_sums_preview":["0","4","20","80","280","896","2688","7680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["Veronese 2 (j=1)"],"edges":["e-4dba393820f57c76"],"text":"A178820 \u2192 Veronese 2 (j=1)"},"headline":"A178820 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-28581f803edaaa59","kind":"fingerprint_related","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":"n-499632d5cca010c1","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-499632d5cca010c1","other":{"id":"n-499632d5cca010c1","name":"veronese_section_2_1 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["8"]],"card_rows":[["2"],["8"],["24","8"],["64","64"]],"row_sums_preview":["0","2","8","32","128","512","2048","8192"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["Veronese 2 (j=1)"],"edges":["e-0d347b78871c0570"],"text":"A038208 \u2192 Veronese 2 (j=1)"},"headline":"A038208 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-36176bebcfebf30a","kind":"fingerprint_related","node_a_id":"n-a75381dbce977094","node_b_id":"n-1f922d5ae3a4fc22","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-a75381dbce977094","other":{"id":"n-a75381dbce977094","name":"veronese_section_2_1 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[[],["-1"],["2"]],"card_rows":[["-1"],["2"],["-3","-1"],["4","4"]],"row_sums_preview":["0","-1","2","-4","8","-16","32","-64"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["Veronese 2 (j=1)"],"edges":["e-a70814c31408551f"],"text":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 (j=1)"},"headline":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-3ab44f1bed44a8b4","kind":"fingerprint_related","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":"n-03178188d92f8622","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-03178188d92f8622","other":{"id":"n-03178188d92f8622","name":"veronese_section_2_1 of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["4"]],"card_rows":[["2"],["4"],["12","4"],["16","16"]],"row_sums_preview":["0","2","4","16","32","128","256","1024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["Veronese 2 (j=1)"],"edges":["e-b898a1f1e7c8e42a"],"text":"A135838 \u2192 Veronese 2 (j=1)"},"headline":"A135838 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-6677b06d3320f9c7","kind":"fingerprint_related","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":"n-29efd6ec0498dec9","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-29efd6ec0498dec9","other":{"id":"n-29efd6ec0498dec9","name":"veronese_section_2_1 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["6"],["42"]],"card_rows":[["6"],["42"],["168","56"],["504","504"]],"row_sums_preview":["0","6","42","224","1008","4032","14784","50688"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["Veronese 2 (j=1)"],"edges":["e-d44b85edec7c5853"],"text":"A178822 \u2192 Veronese 2 (j=1)"},"headline":"A178822 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-9b2c89fb1b628250","kind":"fingerprint_related","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":"n-2e887075d818b756","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-2e887075d818b756","other":{"id":"n-2e887075d818b756","name":"veronese_section_2_1 of binomial","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["2"]],"card_rows":[["1"],["2"],["3","1"],["4","4"]],"row_sums_preview":["0","1","2","4","8","16","32","64"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["Veronese 2 (j=1)"],"edges":["e-3a4f64258abf2a3d"],"text":"A007318 \u2192 Veronese 2 (j=1)"},"headline":"A007318 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-c4f03ab69d6699e2","kind":"fingerprint_related","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":"n-5b51a182e2d7adf3","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-5b51a182e2d7adf3","other":{"id":"n-5b51a182e2d7adf3","name":"veronese_section_2_1 of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["15"],["90"]],"card_rows":[["15"],["90"],["315","105"],["840","840"]],"row_sums_preview":["0","15","90","420","1680","6048","20160","63360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["Veronese 2 (j=1)"],"edges":["e-86ba7e1c6bc69fa7"],"text":"A344565 \u2192 Veronese 2 (j=1)"},"headline":"A344565 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-ce1e8f278393d271","kind":"fingerprint_related","node_a_id":"n-fd133d9d2402ed18","node_b_id":"n-1f922d5ae3a4fc22","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-fd133d9d2402ed18","other":{"id":"n-fd133d9d2402ed18","name":"Triangle of odd-numbered terms in rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["3","1"]],"card_rows":[["1"],["2"],["3","1"],["4","4"]],"row_sums_preview":["1","2","4","8","16","32","64","128"],"regions":["(-inf, 1]"],"region_exact":{},"oeis_ids":["A034867","A202064"],"derivation":{"root":"n-75ccd0453a9b6cec","steps":["x \u21a6 (1/4)x\u22121/4"],"edges":["e-e41ac5ad698885bc"],"text":"A011973 \u2192 x \u21a6 (1/4)x\u22121/4"},"headline":"A034867: Triangle of odd-numbered terms in rows of Pascal's triangle.","chain":"A011973 \u2192 x \u21a6 (1/4)x\u22121/4"}},{"id":"l-ee41c99dcecf3d18","kind":"fingerprint_related","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":"n-c3aceb3db6a31013","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-c3aceb3db6a31013","other":{"id":"n-c3aceb3db6a31013","name":"veronese_section_2_1 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["4"]],"card_rows":[["2"],["4"],["6","2"],["8","8"]],"row_sums_preview":["0","2","4","8","16","32","64","128"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["Veronese 2 (j=1)"],"edges":["e-53a911614b54e844"],"text":"A028326 \u2192 Veronese 2 (j=1)"},"headline":"A028326 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-fce667a437d674a7","kind":"fingerprint_related","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":"n-0ed898288ff872d1","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-0ed898288ff872d1","other":{"id":"n-0ed898288ff872d1","name":"veronese_section_2_1 of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["6"],["12","4"],["20","20"]],"row_sums_preview":["0","0","2","6","16","40","96","224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["D","Veronese 2 (j=1)"],"edges":["e-fd16b43297f2bb3e","e-e2abcf6e1ca1607c"],"text":"A007318 \u2192 D \u2192 Veronese 2 (j=1)"},"headline":"A007318 \u2192 D \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-0902ede98a26bf75","kind":"oeis_match","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":null,"fingerprint":null,"oeis_id":"A034008","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A034008","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-5e046abbeff76e93","kind":"oeis_match","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":null,"fingerprint":null,"oeis_id":"A190126","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":14},"in_catalog":false,"oeis_id":"A190126","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-80f1248ce04461c3","kind":"oeis_match","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":null,"fingerprint":null,"oeis_id":"A011782","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":1,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A011782","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-8ded81d1ba42e1e9","kind":"oeis_match","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":null,"fingerprint":null,"oeis_id":"A131577","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A131577","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-9c4f32246c519655","kind":"oeis_match","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":null,"fingerprint":null,"oeis_id":"A034867","in_catalog":1,"target":"triangle","row_alignment":"node_lengths","rows_matched":14,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":true,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":14,"target":"triangle","terms_compared":56},"in_catalog":true,"oeis_id":"A034867","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a115d190ea1ddc9e","kind":"oeis_match","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":null,"fingerprint":null,"oeis_id":"A166444","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":14},"in_catalog":false,"oeis_id":"A166444","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a326925ab08c83db","kind":"oeis_match","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":null,"fingerprint":null,"oeis_id":"A000079","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":14},"in_catalog":false,"oeis_id":"A000079","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b2fd1d05a5bb3cf9","kind":"oeis_match","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":null,"fingerprint":null,"oeis_id":"A050732","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":14},"in_catalog":false,"oeis_id":"A050732","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d2068f0dee7e40c6","kind":"oeis_match","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":null,"fingerprint":null,"oeis_id":"A109447","in_catalog":1,"target":"reversed_triangle","row_alignment":"node_lengths","rows_matched":14,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":true,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":14,"target":"reversed_triangle","terms_compared":56},"in_catalog":true,"oeis_id":"A109447","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f3e4f4fa1641d6be","kind":"oeis_match","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":null,"fingerprint":null,"oeis_id":"A118655","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":14},"in_catalog":false,"oeis_id":"A118655","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A109447","relation":"match_reversed_triangle"},{"oeis_id":"A000079","relation":"match_row_sums"},{"oeis_id":"A011782","relation":"match_row_sums"},{"oeis_id":"A034008","relation":"match_row_sums"},{"oeis_id":"A050732","relation":"match_row_sums"},{"oeis_id":"A118655","relation":"match_row_sums"},{"oeis_id":"A131577","relation":"match_row_sums"},{"oeis_id":"A166444","relation":"match_row_sums"},{"oeis_id":"A190126","relation":"match_row_sums"},{"oeis_id":"A034867","relation":"match_triangle"}],"bfiles":[{"node_id":"n-1f922d5ae3a4fc22","oeis_id":"A000079","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":35,"oeis_terms_available":35,"term_count":200,"raw_size":6981},{"node_id":"n-1f922d5ae3a4fc22","oeis_id":"A034867","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":62,"terms_matched":56,"oeis_terms_agreeing":64,"oeis_terms_available":64,"term_count":992,"raw_size":13447},{"node_id":"n-1f922d5ae3a4fc22","oeis_id":"A109447","target":"reversed_triangle","oeis_offset":1,"first_row":1,"rows_used":62,"terms_matched":56,"oeis_terms_agreeing":64,"oeis_terms_available":64,"term_count":992,"raw_size":13449},{"node_id":"n-1f922d5ae3a4fc22","oeis_id":"A131577","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":35,"oeis_terms_available":35,"term_count":200,"raw_size":6922}],"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: 0;\n  1: 0, 1;\n  2: 0, 2;\n  3: 0, 3,  1;\n  4: 0, 4,  4;\n  5: 0, 5, 10,   1;\n  6: 0, 6, 20,   6;\n  7: 0, 7, 35,  21,  1;\n  8: 0, 8, 56,  56,  8;\n  9: 0, 9, 84, 126, 36, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = 0, P(1,x) = x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - x)*v[i - 2]); 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}},{"oeis_id":"A034867","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"low","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  2;\n  2:  3,   1;\n  3:  4,   4;\n  4:  5,  10,   1;\n  5:  6,  20,   6;\n  6:  7,  35,  21,   1;\n  7:  8,  56,  56,   8;\n  8:  9,  84, 126,  36,  1;\n  9: 10, 120, 252, 120, 10;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = 2*x.","mathematica":"P[0, x_] := x;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 2, x]];\nTable[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - x)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":"low","regular_triangle":false}},{"oeis_id":"A109447","target":"reversed_triangle","first_row":1,"oeis_offset":1,"reversed":true,"trim_zeros":"low","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2:  2;\n   3:  1,   3;\n   4:  4,   4;\n   5:  1,  10,   5;\n   6:  6,  20,   6;\n   7:  1,  21,  35,   7;\n   8:  8,  56,  56,   8;\n   9:  1,  36, 126,  84,  9;\n  10: 10, 120, 252, 120, 10;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) for n >= 3, with P(1,x) = x, P(2,x) = 2*x.","mathematica":"P[1, x_] := x;\nP[2, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 2, x]];\nTable[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = x; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = 2*v[i - 1] - (1 - x)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":true,"trim_zeros":"low","regular_triangle":false}}],"derivation":{"root":"n-7717563017cb9e5f","steps":["Veronese 2 (j=1)"],"edges":["e-6fdf1329e5e61ed8"],"text":"A159854 \u2192 Veronese 2 (j=1)"},"headline":"A159854 \u2192 Veronese 2 (j=1)","chain":null}