{"id":"n-1864d83319c9138c","revision":1,"first_index":0,"fingerprint":"f6-ce65a3e6fcbea790","name":"veronese_section_2_1 of Triangle of coefficients in expansion of (1+2*x)^n.","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":[[],["2"],["4"],["6","8"],["8","32"],["10","80","32"],["12","160","192"],["14","280","672","128"],["16","448","1792","1024"],["18","672","4032","4608","512"],["20","960","8064","15360","5120"],["22","1320","14784","42240","28160","2048"],["24","1760","25344","101376","112640","24576"],["26","2288","41184","219648","366080","159744","8192"],["28","2912","64064","439296","1025024","745472","114688"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","2","4","14","40","122","364","1094","3280","9842","29524","88574","265720","797162","2391484"],"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"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163281,"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-c6792048648d3d6d"]},{"id":163282,"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-c6792048648d3d6d"]},{"id":163283,"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-c6792048648d3d6d"]}],"region_exact":{},"incoming_edges":[{"id":"e-c6792048648d3d6d","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-1864d83319c9138c","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":"46d78ec19c7a8c03ad5b8a4e00e0cf6a54d48da30451a9ec4409ba091a986cef","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-71965e79e47a4526","role":"sequence","name":"Triangle of coefficients in expansion of (1+2*x)^n.","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-0846f19d4cae7d38","kind":"fingerprint_related","node_a_id":"n-cf38558d99f4fe37","node_b_id":"n-1864d83319c9138c","fingerprint":"f6-ce65a3e6fcbea790","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ce65a3e6fcbea790"},"other_id":"n-cf38558d99f4fe37","other":{"id":"n-cf38558d99f4fe37","name":"Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).","first_index":0,"fingerprint":"f6-ce65a3e6fcbea790","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["3","4"]],"card_rows":[["1"],["2"],["3","4"],["4","16"]],"row_sums_preview":["1","2","7","20","61","182","547","1640"],"regions":["(-inf, 1/4]","(-inf, 0]"],"region_exact":{},"oeis_ids":["A229032"],"derivation":{"root":"n-75ccd0453a9b6cec","steps":["x \u21a6 x\u22121/4"],"edges":["e-44df003225948a93"],"text":"A011973 \u2192 x \u21a6 x\u22121/4"},"headline":"A229032: Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).","chain":"A011973 \u2192 x \u21a6 x\u22121/4"}},{"id":"l-147b1ad2cb8d1416","kind":"fingerprint_related","node_a_id":"n-1864d83319c9138c","node_b_id":"n-82e38e2652e8d358","fingerprint":"f6-ce65a3e6fcbea790","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ce65a3e6fcbea790"},"other_id":"n-82e38e2652e8d358","other":{"id":"n-82e38e2652e8d358","name":"veronese_section_2_1 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-ce65a3e6fcbea790","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["6"],["12"]],"card_rows":[["6"],["12"],["18","24"],["24","96"]],"row_sums_preview":["0","6","12","42","120","366","1092","3282"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["Veronese 2 (j=1)"],"edges":["e-1dcfbe1f89fd593f"],"text":"A120909 \u2192 Veronese 2 (j=1)"},"headline":"A120909 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-9871d48336e456b6","kind":"oeis_match","node_a_id":"n-1864d83319c9138c","node_b_id":null,"fingerprint":null,"oeis_id":"A152011","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":"A152011","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A152011","relation":"match_row_sums"}],"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:  0;\n  1:  2;\n  2:  4;\n  3:  6,   8;\n  4:  8,  32;\n  5: 10,  80,   32;\n  6: 12, 160,  192;\n  7: 14, 280,  672,  128;\n  8: 16, 448, 1792, 1024;\n  9: 18, 672, 4032, 4608, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-4*x)*P(n-2,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - 4*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - 4*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}}],"derivation":{"root":"n-71965e79e47a4526","steps":["Veronese 2 (j=1)"],"edges":["e-c6792048648d3d6d"],"text":"A013609 \u2192 Veronese 2 (j=1)"},"headline":"A013609 \u2192 Veronese 2 (j=1)","chain":null}