{"id":"n-66ad0c588cd0c2e0","revision":0,"first_index":0,"fingerprint":"f6-702b8764f2c773f5","name":"shift_x_plus_one of veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["8","4"],["32","24"],["128","128","16"],["512","640","160"],["2048","3072","1152","64"],["8192","14336","7168","896"],["32768","65536","40960","8192","256"],["131072","294912","221184","61440","4608"],["524288","1310720","1146880","409600","51200","1024"],["2097152","5767168","5767168","2523136","450560","22528"],["8388608","25165824","28311552","14680064","3440640","294912","4096"],["33554432","109051904","136314880","81788928","23855104","2981888","106496"],["134217728","469762048","645922816","440401920","154140672","25690112","1605632","16384"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","12","56","272","1312","6336","30592","147712","713216","3443712","16627712","80285696","387653632","1871757312"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-4","-2"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-9f9e3432a5bc70e2","recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176213,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-6a9b740d2b798c13"]},{"id":176214,"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-4c7f7e6e93d0f4c2","e-6a9b740d2b798c13"]},{"id":176215,"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-4c7f7e6e93d0f4c2","e-6a9b740d2b798c13"]},{"id":176216,"position":3,"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-4c7f7e6e93d0f4c2"]},{"id":176217,"position":4,"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-4c7f7e6e93d0f4c2"]}],"region_exact":{},"incoming_edges":[{"id":"e-4c7f7e6e93d0f4c2","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-66ad0c588cd0c2e0","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":"c82792f3d01ef0ac487affc092f91c5545868824cfea7a019e1d05efb71af09e","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-1b9fc04e4a40039e","role":"sequence","name":"veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","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]"}]},{"id":"e-9f9e3432a5bc70e2","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-66ad0c588cd0c2e0","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"0053024ac6db1ad9cfcf600a76f2aad98c8f29b3114b1a53087f6dc089179665","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-6a9b740d2b798c13","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-66ad0c588cd0c2e0","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"d71b38baf715ebb06caa29a318438d271959570064e92087e82057d632ec0e83","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-1b9fc04e4a40039e","role":"sequence","name":"veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","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-0cf59192a05e5035","kind":"fingerprint_related","node_a_id":"n-0d4f187a7cfebed3","node_b_id":"n-66ad0c588cd0c2e0","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-0d4f187a7cfebed3","other":{"id":"n-0d4f187a7cfebed3","name":"shift_x_plus_one of veronese_section_2_0 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["2"],["4","2"]],"card_rows":[["2"],["2"],["4","2"],["8","6"]],"row_sums_preview":["2","2","6","14","34","82","198","478"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-e811ab86eff6d185","e-1f4bc87c26595050"],"text":"A028326 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A028326 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-21a0b233942ac987","kind":"fingerprint_related","node_a_id":"n-4d1e2be69e19d889","node_b_id":"n-66ad0c588cd0c2e0","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-4d1e2be69e19d889","other":{"id":"n-4d1e2be69e19d889","name":"mobius_x_over_x_plus_one of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).","first_index":1,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","1","2"]],"card_rows":[["1"],["0","1"],["0","1","2"],["0","0","3","4"]],"row_sums_preview":["1","1","3","7","17","41","99","239"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf23198864b5678a","steps":["x \u21a6 x/(x+1)"],"edges":["e-c830e34032c85a28"],"text":"A098158 \u2192 x \u21a6 x/(x+1)"},"headline":"A098158 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-4555b69ee39f2d8d","kind":"fingerprint_related","node_a_id":"n-cd85e79f8f3e8a20","node_b_id":"n-66ad0c588cd0c2e0","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-cd85e79f8f3e8a20","other":{"id":"n-cd85e79f8f3e8a20","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of circular binary words of length n having k occurrences of 01 (0 <= k <= floor(n/2)).","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["4","2"]],"card_rows":[["1"],["2"],["4","2"],["8","6"]],"row_sums_preview":["1","2","6","14","34","82","198","478"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ced86a7ce48c966e","steps":["basis A007318"],"edges":["e-6dc4734e89bdcd0a"],"text":"A119462 \u2192 basis A007318"},"headline":"A119462 \u2192 basis A007318","chain":null}},{"id":"l-7b479b74792d4f20","kind":"fingerprint_related","node_a_id":"n-66ad0c588cd0c2e0","node_b_id":"n-d7283fa0e9ac22b4","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-d7283fa0e9ac22b4","other":{"id":"n-d7283fa0e9ac22b4","name":"shift_x_plus_one of Table: T(n,k) = n!*binomial(n+1,2*k).","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["8","6"]],"card_rows":[["1"],["2","1"],["8","6"],["48","48","6"]],"row_sums_preview":["1","3","14","102","984","11880","172080","2908080"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-90f6949024558b8d","steps":["basis A007318"],"edges":["e-26a3e60cc630357d"],"text":"A228955 \u2192 basis A007318"},"headline":"A228955 \u2192 basis A007318","chain":null}},{"id":"l-7c688049f0d4aff4","kind":"fingerprint_related","node_a_id":"n-5b714ee1f71b153f","node_b_id":"n-66ad0c588cd0c2e0","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-5b714ee1f71b153f","other":{"id":"n-5b714ee1f71b153f","name":"shift_x_plus_one of veronese_section_2_0 of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-702b8764f2c773f5","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"],["6","3"],["16","12"]],"row_sums_preview":["0","1","2","9","28","85","246","693"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-30ed1b2c5535cdc0","e-0b9e68fa7443398c"],"text":"A003506 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A003506 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-7d883061fd2d0b2f","kind":"fingerprint_related","node_a_id":"n-66ad0c588cd0c2e0","node_b_id":"n-f47ef477093d1825","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-f47ef477093d1825","other":{"id":"n-f47ef477093d1825","name":"shift_x_plus_one of veronese_section_2_0 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6"],["42","21"]],"card_rows":[["1"],["6"],["42","21"],["224","168"]],"row_sums_preview":["1","6","63","392","2142","10332","45738","189288"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-496b1966a8762df1","e-3dd245e74534b1c0"],"text":"A178822 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A178822 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-7f2d331b512eca14","kind":"fingerprint_related","node_a_id":"n-66ad0c588cd0c2e0","node_b_id":"n-fa2ea18f13501719","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-fa2ea18f13501719","other":{"id":"n-fa2ea18f13501719","name":"shift_x_plus_one of veronese_section_2_0 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5"],["30","15"]],"card_rows":[["1"],["5"],["30","15"],["140","105"]],"row_sums_preview":["1","5","45","245","1190","5166","20790","78870"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-8ba085c2fb60bb1b","e-68de572e63ababc1"],"text":"A178821 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A178821 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-a1559b8f625a7d4f","kind":"fingerprint_related","node_a_id":"n-97deebd28987ca5a","node_b_id":"n-66ad0c588cd0c2e0","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-97deebd28987ca5a","other":{"id":"n-97deebd28987ca5a","name":"shift_x_plus_one of veronese_section_2_0 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4"],["20","10"]],"card_rows":[["1"],["4"],["20","10"],["80","60"]],"row_sums_preview":["1","4","30","140","595","2296","8316","28680"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["Veronese 2","basis A007318"],"edges":["e-9985dc3bbd6a1733","e-05c29514f8b31a01"],"text":"A178820 \u2192 Veronese 2 \u2192 basis A007318"},"headline":"A178820 \u2192 Veronese 2 \u2192 basis A007318","chain":null}},{"id":"l-b27ff1a70f9cf6d1","kind":"fingerprint_related","node_a_id":"n-495335f6a4ca3289","node_b_id":"n-66ad0c588cd0c2e0","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-495335f6a4ca3289","other":{"id":"n-495335f6a4ca3289","name":"Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["4","3"]],"row_sums_preview":["1","1","3","7","17","41","99","239"],"regions":["(-inf, 0)","(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":["A201701"],"derivation":{"root":"n-495335f6a4ca3289","steps":[],"edges":[],"text":"A201701"},"headline":"A201701: Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).","chain":null}},{"id":"l-f018076f044dac8b","kind":"fingerprint_related","node_a_id":"n-777834828b7583c2","node_b_id":"n-66ad0c588cd0c2e0","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-777834828b7583c2","other":{"id":"n-777834828b7583c2","name":"shift_x_plus_one of veronese_section_2_0 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-1"],["2","1"]],"card_rows":[["1"],["-1"],["2","1"],["-4","-3"]],"row_sums_preview":["1","-1","3","-7","17","-41","99","-239"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-969b818b29eb86a6","e-9f1fd6b6bc67e5d0"],"text":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 \u2192 x \u21a6 x+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:      2;\n  2:      8,      4;\n  3:     32,     24;\n  4:    128,    128,     16;\n  5:    512,    640,    160;\n  6:   2048,   3072,   1152,    64;\n  7:   8192,  14336,   7168,   896;\n  8:  32768,  65536,  40960,  8192,  256;\n  9: 131072, 294912, 221184, 61440, 4608;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (4*n-4+(2*n-2)*x)*P(n-1,x) - (4*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (2*n - 2)*x)*P[n - 1, x] - (4*x + 4*x^2)*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (2*n - 2)*x)*v[i - 1] - (4*x + 4*x^2)*deriv(v[i - 1], x))/(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-d28ad9841fc73317","steps":["Veronese 2","basis A007318"],"edges":["e-01a436d0e1f57933","e-4c7f7e6e93d0f4c2"],"text":"A038208 \u2192 Veronese 2 \u2192 basis A007318"},"headline":"A038208 \u2192 Veronese 2 \u2192 basis A007318","chain":null}