{"id":"n-516f4a2401b3e46a","revision":0,"first_index":0,"fingerprint":"f6-8b04bb8c3994e856","name":"shift_x_plus_one of 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"],["14","8"],["40","32"],["122","144","32"],["364","544","192"],["1094","2008","1056","128"],["3280","7104","4864","1024"],["9842","24608","20928","6656","512"],["29524","83648","84864","35840","5120"],["88574","280488","330944","175360","38400","2048"],["265720","930016","1251072","797696","235520","24576"],["797162","3055792","4616928","3445248","1287680","208896","8192"],["2391484","9964512","16707136","14287872","6472704","1433600","114688"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","2","4","22","72","298","1100","4286","16272","62546","238996","915814","3504600","13419898","51371996"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-8","-8"],["5","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","-14","-8"]]},"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-a3a439732d4e9f75","recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173616,"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-748cbf1d594485ee"]},{"id":173617,"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-64918b70a67c9b9b","e-748cbf1d594485ee"]},{"id":173618,"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-64918b70a67c9b9b","e-748cbf1d594485ee"]},{"id":173619,"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-64918b70a67c9b9b"]},{"id":173620,"position":4,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-64918b70a67c9b9b"]}],"region_exact":{},"incoming_edges":[{"id":"e-64918b70a67c9b9b","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-516f4a2401b3e46a","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":"64d32886b9e1a9648f9aa1c6cbe828bf1d8dc68d34c9f76db7bf9d134640812f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-1864d83319c9138c","role":"sequence","name":"veronese_section_2_1 of 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]"}]},{"id":"e-a3a439732d4e9f75","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-516f4a2401b3e46a","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":"2c81796a759bc26d8ba91869b36327ba4901510f7ba6a246f42c7ce6b5f62427","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-748cbf1d594485ee","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-516f4a2401b3e46a","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":"7649c3228fe45e8e5bf65d241eab7e1caa2efa571ba689a51d6a321a9cacf427","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-1864d83319c9138c","role":"sequence","name":"veronese_section_2_1 of 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-25103f700e44f4bd","kind":"fingerprint_related","node_a_id":"n-516f4a2401b3e46a","node_b_id":"n-e74cbe51bed5cf8e","fingerprint":"f6-8b04bb8c3994e856","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8b04bb8c3994e856"},"other_id":"n-e74cbe51bed5cf8e","other":{"id":"n-e74cbe51bed5cf8e","name":"shift_x_plus_one of 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":"f6-8b04bb8c3994e856","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"],["42","24"],["120","96"]],"row_sums_preview":["0","6","12","66","216","894","3300","12858"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["Veronese 2 (j=1)","x \u21a6 x+1"],"edges":["e-1dcfbe1f89fd593f","e-8bc1db2bd6725c95"],"text":"A120909 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1"},"headline":"A120909 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1","chain":null}},{"id":"l-e5ebc0646ad31458","kind":"fingerprint_related","node_a_id":"n-e5bdf85b655e989a","node_b_id":"n-516f4a2401b3e46a","fingerprint":"f6-8b04bb8c3994e856","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8b04bb8c3994e856"},"other_id":"n-e5bdf85b655e989a","other":{"id":"n-e5bdf85b655e989a","name":"shift_x_plus_one of 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":"f6-8b04bb8c3994e856","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["7","4"]],"card_rows":[["1"],["2"],["7","4"],["20","16"]],"row_sums_preview":["1","2","11","36","149","550","2143","8136"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf38558d99f4fe37","steps":["x \u21a6 x+1"],"edges":["e-7b35457a4f9e8cf8"],"text":"A229032 \u2192 x \u21a6 x+1"},"headline":"A229032 \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:    0;\n  1:    2;\n  2:    4;\n  3:   14,     8;\n  4:   40,    32;\n  5:  122,   144,    32;\n  6:  364,   544,   192;\n  7: 1094,  2008,  1056,  128;\n  8: 3280,  7104,  4864, 1024;\n  9: 9842, 24608, 20928, 6656, 512;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (5*n-8+(4*n-8)*x)*P(n-1,x) - (6+14*x+8*x^2)*d/dx P(n-1,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[Together[((5*n - 8 + (4*n - 8)*x)*P[n - 1, x] - (6 + 14*x + 8*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] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((5*n - 8 + (4*n - 8)*x)*v[i - 1] - (6 + 14*x + 8*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-71965e79e47a4526","steps":["Veronese 2 (j=1)","basis A007318"],"edges":["e-c6792048648d3d6d","e-64918b70a67c9b9b"],"text":"A013609 \u2192 Veronese 2 (j=1) \u2192 basis A007318"},"headline":"A013609 \u2192 Veronese 2 (j=1) \u2192 basis A007318","chain":null}