{"id":"n-46e97bc7afa673ad","revision":0,"first_index":0,"fingerprint":"f6-97ff5e3c0ee49bcd","name":"shift_x_plus_one of veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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"],["5","1"],["14","6"],["41","26","1"],["122","100","10"],["365","363","63","1"],["1094","1274","322","14"],["3281","4372","1462","116","1"],["9842","14760","6156","744","18"],["29525","49205","24610","4090","185","1"],["88574","162382","94732","20284","1430","22"],["265721","531438","354303","93476","9255","270","1"],["797162","1727180","1295398","407784","53014","2444","26"],["2391485","5580127","4650121","1705011","277767","18221","371","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","6","20","68","232","792","2704","9232","31520","107616","367424","1254464","4283008","14623104"],"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":[["-2"],["2"]]},"terms":[{"coeff":{"coeffs":[["-5","-1"],["5","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["6","4","-2"]]},"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-744b7f8ea783d6f3","recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173861,"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-5eac55e2914f1c1a"]},{"id":173862,"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-5eac55e2914f1c1a","e-9df69e49144f88d9"]},{"id":173863,"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-5eac55e2914f1c1a","e-9df69e49144f88d9"]},{"id":173864,"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-9df69e49144f88d9"]},{"id":173865,"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-9df69e49144f88d9"]}],"region_exact":{},"incoming_edges":[{"id":"e-9df69e49144f88d9","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-46e97bc7afa673ad","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":"a9fade0bef29f7a54eed1233b3dbdfee4afbb3a5cc73a0633315c2d86a0f149e","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-b6912a345c4fb865","role":"sequence","name":"veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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-744b7f8ea783d6f3","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-46e97bc7afa673ad","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":"118aa3a1e9bec8ad99d2f0b2f2c1637cd683cb152e7bd1d7efe70f5546d2ef5c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-5eac55e2914f1c1a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-46e97bc7afa673ad","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":"39eb0f7fe27333a50187a71d7a9725016fadf4489e16cc1dd672a080ca19071f","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-b6912a345c4fb865","role":"sequence","name":"veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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-3477c37490f35b91","kind":"fingerprint_related","node_a_id":"n-048e7c9e10af99dd","node_b_id":"n-46e97bc7afa673ad","fingerprint":"f6-97ff5e3c0ee49bcd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-97ff5e3c0ee49bcd"},"other_id":"n-048e7c9e10af99dd","other":{"id":"n-048e7c9e10af99dd","name":"Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","first_index":1,"fingerprint":"f6-97ff5e3c0ee49bcd","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["5","1"]],"card_rows":[["1"],["2"],["5","1"],["14","6"]],"row_sums_preview":["1","2","6","20","68","232","792","2704"],"regions":["(-inf, 0)"],"region_exact":{},"oeis_ids":["A201730"],"derivation":{"root":"n-048e7c9e10af99dd","steps":[],"edges":[],"text":"A201730"},"headline":"A201730: Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","chain":null}},{"id":"l-7352d0dfbf128813","kind":"fingerprint_related","node_a_id":"n-46e97bc7afa673ad","node_b_id":"n-2ca20f5a0bd85ef7","fingerprint":"f6-97ff5e3c0ee49bcd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-97ff5e3c0ee49bcd"},"other_id":"n-2ca20f5a0bd85ef7","other":{"id":"n-2ca20f5a0bd85ef7","name":"shift_x_plus_one of veronese_section_2_0 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":"f6-97ff5e3c0ee49bcd","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3"],["6"],["15","3"]],"card_rows":[["3"],["6"],["15","3"],["42","18"]],"row_sums_preview":["3","6","18","60","204","696","2376","8112"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-74cdb64b6d1e6b98","e-4ea4269ef8cba4ff"],"text":"A120910 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A120910 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-8b2bc58c7eb4ab0c","kind":"fingerprint_related","node_a_id":"n-7556acd10c14a2be","node_b_id":"n-46e97bc7afa673ad","fingerprint":"f6-97ff5e3c0ee49bcd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-97ff5e3c0ee49bcd"},"other_id":"n-7556acd10c14a2be","other":{"id":"n-7556acd10c14a2be","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of descents.","first_index":0,"fingerprint":"f6-97ff5e3c0ee49bcd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","2"],["0","1","5"]],"card_rows":[["1"],["0","2"],["0","1","5"],["0","0","6","14"]],"row_sums_preview":["1","2","6","20","68","232","792","2704"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5576233c61c40153","steps":["x \u21a6 x/(x+1)"],"edges":["e-07f697f587045455"],"text":"A372873 \u2192 x \u21a6 x/(x+1)"},"headline":"A372873 \u2192 x \u21a6 x/(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:    5,     1;\n  3:   14,     6;\n  4:   41,    26,    1;\n  5:  122,   100,   10;\n  6:  365,   363,   63,   1;\n  7: 1094,  1274,  322,  14;\n  8: 3281,  4372, 1462, 116,  1;\n  9: 9842, 14760, 6156, 744, 18;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = (5*n-5+(n-1)*x)*P(n-1,x) + (6+4*x-2*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[((5*n - 5 + (n - 1)*x)*P[n - 1, x] + (6 + 4*x - 2*x^2)*D[P[n - 1, x], x])/(2*n - 2)]];\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] = ((5*n - 5 + (n - 1)*x)*v[i - 1] + (6 + 4*x - 2*x^2)*deriv(v[i - 1], x))/(2*n - 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-b6912a345c4fb865","steps":["x \u21a6 x+1"],"edges":["e-5eac55e2914f1c1a"],"text":"A372868 \u2192 x \u21a6 x+1"},"headline":"A372868 \u2192 x \u21a6 x+1","chain":null}
