{"id":"n-a8ee0944a0199e83","revision":1,"first_index":0,"fingerprint":"f6-3b37e384e5c5a5b6","name":"veronese_section_2_0 of Even-numbered rows of Pascal's triangle.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","1"],["1","6","1"],["1","15","15","1"],["1","28","70","28","1"],["1","45","210","210","45","1"],["1","66","495","924","495","66","1"],["1","91","1001","3003","3003","1001","91","1"],["1","120","1820","8008","12870","8008","1820","120","1"],["1","153","3060","18564","43758","43758","18564","3060","153","1"],["1","190","4845","38760","125970","184756","125970","38760","4845","190","1"],["1","231","7315","74613","319770","646646","646646","319770","74613","7315","231","1"],["1","276","10626","134596","735471","1961256","2704156","1961256","735471","134596","10626","276","1"],["1","325","14950","230230","1562275","5311735","9657700","9657700","5311735","1562275","230230","14950","325","1"],["1","378","20475","376740","3108105","13123110","30421755","40116600","30421755","13123110","3108105","376740","20475","378","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","8","32","128","512","2048","8192","32768","131072","524288","2097152","8388608","33554432","134217728"],"nonn":true,"definitions":[{"position":0,"source":"derived:veronese_section_2_0","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168439,"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-f72875fb5d524bf0"]},{"id":168440,"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-f72875fb5d524bf0"]},{"id":168441,"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-f72875fb5d524bf0"]}],"region_exact":{},"incoming_edges":[{"id":"e-f72875fb5d524bf0","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-a8ee0944a0199e83","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":"b18e4e976fc2addb0c5c6605366822c014e091e64b1c2f2d4461304576481a07","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-08f49539099949d5","role":"sequence","name":"Even-numbered rows of Pascal's triangle.","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-a6685b47e8dc458f","kind":"fingerprint_related","node_a_id":"n-f7935c36a3c0ec8c","node_b_id":"n-a8ee0944a0199e83","fingerprint":"f6-3b37e384e5c5a5b6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3b37e384e5c5a5b6"},"other_id":"n-f7935c36a3c0ec8c","other":{"id":"n-f7935c36a3c0ec8c","name":"Triangle read by rows: T(n, k) = binomial(2n, 2k).","first_index":1,"fingerprint":"f6-3b37e384e5c5a5b6","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","6","1"]],"card_rows":[["1"],["1","1"],["1","6","1"],["1","15","15","1"]],"row_sums_preview":["1","2","8","32","128","512","2048","8192"],"regions":["(-inf, +inf)","(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A086645"],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":[],"edges":[],"text":"A086645"},"headline":"A086645: Triangle read by rows: T(n, k) = binomial(2n, 2k).","chain":null}},{"id":"l-1493dd758421efa6","kind":"oeis_match","node_a_id":"n-a8ee0944a0199e83","node_b_id":null,"fingerprint":null,"oeis_id":"A081654","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A081654","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-3cd7ed83a2358b2c","kind":"oeis_match","node_a_id":"n-a8ee0944a0199e83","node_b_id":null,"fingerprint":null,"oeis_id":"A004171","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":"A004171","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-449f2538efafef21","kind":"oeis_match","node_a_id":"n-a8ee0944a0199e83","node_b_id":null,"fingerprint":null,"oeis_id":"A081294","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":"A081294","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-77536385e004a5a4","kind":"oeis_match","node_a_id":"n-a8ee0944a0199e83","node_b_id":null,"fingerprint":null,"oeis_id":"A086645","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":59},"in_catalog":true,"oeis_id":"A086645","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-900f074016202f17","kind":"oeis_match","node_a_id":"n-a8ee0944a0199e83","node_b_id":null,"fingerprint":null,"oeis_id":"A086645","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":59},"in_catalog":true,"oeis_id":"A086645","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A086645","relation":"match_reversed_triangle"},{"oeis_id":"A004171","relation":"match_row_sums"},{"oeis_id":"A081294","relation":"match_row_sums"},{"oeis_id":"A081654","relation":"match_row_sums"},{"oeis_id":"A086645","relation":"match_triangle"}],"bfiles":[{"node_id":"n-a8ee0944a0199e83","oeis_id":"A004171","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":13031},{"node_id":"n-a8ee0944a0199e83","oeis_id":"A081294","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":26,"oeis_terms_available":26,"term_count":200,"raw_size":12911},{"node_id":"n-a8ee0944a0199e83","oeis_id":"A086645","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":59,"oeis_terms_agreeing":59,"oeis_terms_available":59,"term_count":990,"raw_size":16483},{"node_id":"n-a8ee0944a0199e83","oeis_id":"A086645","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":59,"oeis_terms_agreeing":59,"oeis_terms_available":59,"term_count":990,"raw_size":16483}],"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,   1;\n  2: 1,   6,    1;\n  3: 1,  15,   15,     1;\n  4: 1,  28,   70,    28,     1;\n  5: 1,  45,  210,   210,    45,     1;\n  6: 1,  66,  495,   924,   495,    66,     1;\n  7: 1,  91, 1001,  3003,  3003,  1001,    91,    1;\n  8: 1, 120, 1820,  8008, 12870,  8008,  1820,  120,   1;\n  9: 1, 153, 3060, 18564, 43758, 43758, 18564, 3060, 153, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1-2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 2*x + x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 - 2*x + x^2)*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":"A086645","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,   1;\n  2: 1,   6,    1;\n  3: 1,  15,   15,     1;\n  4: 1,  28,   70,    28,     1;\n  5: 1,  45,  210,   210,    45,     1;\n  6: 1,  66,  495,   924,   495,    66,     1;\n  7: 1,  91, 1001,  3003,  3003,  1001,    91,    1;\n  8: 1, 120, 1820,  8008, 12870,  8008,  1820,  120,   1;\n  9: 1, 153, 3060, 18564, 43758, 43758, 18564, 3060, 153, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1-2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 2*x + x^2)*P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 - 2*x + x^2)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}},{"oeis_id":"A086645","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,   1;\n  2: 1,   6,    1;\n  3: 1,  15,   15,     1;\n  4: 1,  28,   70,    28,     1;\n  5: 1,  45,  210,   210,    45,     1;\n  6: 1,  66,  495,   924,   495,    66,     1;\n  7: 1,  91, 1001,  3003,  3003,  1001,    91,    1;\n  8: 1, 120, 1820,  8008, 12870,  8008,  1820,  120,   1;\n  9: 1, 153, 3060, 18564, 43758, 43758, 18564, 3060, 153, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*x)*P(n-1,x) - (1-2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 2*x + x^2)*P[n - 2, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 - 2*x + x^2)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-08f49539099949d5","steps":["Veronese 2"],"edges":["e-f72875fb5d524bf0"],"text":"A034870 \u2192 Veronese 2"},"headline":"A034870 \u2192 Veronese 2","chain":null}
