{"id":"n-1e7342ec7a250847","revision":1,"first_index":0,"fingerprint":"f6-24cf4d809f9b6fe7","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).","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"],["1"],["3","2"],["7","6"],["19","24","6"],["51","80","30"],["141","270","150","20"],["393","882","630","140"],["1107","2856","2520","840","70"],["3139","9144","9576","4200","630"],["8953","29070","35280","19320","4410","252"],["25653","91850","126720","83160","25410","2772"],["73789","288816","446490","341880","131670","22176","924"],["212941","904488","1548690","1355640","630630","144144","12012"],["616227","2823002","5303298","5225220","2852850","828828","108108","3432"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","5","13","49","161","581","2045","7393","26689","97285","355565","1305745","4808545","17760965"],"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"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-4","-4"],["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"}}],"properties":[{"id":163578,"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-0f1b9062159c7e91"]},{"id":163579,"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-0f1b9062159c7e91"]},{"id":163580,"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-0f1b9062159c7e91"]}],"region_exact":{},"incoming_edges":[{"id":"e-0f1b9062159c7e91","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-1e7342ec7a250847","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":"7b803dea0ae292215c9d84a1c83c837d0258d6985a383424288ea6610362cca2","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-8bfdd58279be42d7","role":"sequence","name":"Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).","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-aa3108fe4d7a56cf","kind":"fingerprint_related","node_a_id":"n-123dac0e8ec7cfb1","node_b_id":"n-1e7342ec7a250847","fingerprint":"f6-24cf4d809f9b6fe7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-24cf4d809f9b6fe7"},"other_id":"n-123dac0e8ec7cfb1","other":{"id":"n-123dac0e8ec7cfb1","name":"shift_x_plus_one of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.","first_index":1,"fingerprint":"f6-24cf4d809f9b6fe7","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["3","2"]],"card_rows":[["1"],["1"],["3","2"],["7","6"]],"row_sums_preview":["1","1","5","13","49","161","581","2045"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c914b69838ac70ec","steps":["\u2299 1/k! (shift -1)","x \u21a6 x+1"],"edges":["e-4d3032ecd7423900","e-502ba5da9494e9e3"],"text":"A059344 \u2192 \u2299 1/k! (shift -1) \u2192 x \u21a6 x+1"},"headline":"A059344 \u2192 \u2299 1/k! (shift -1) \u2192 x \u21a6 x+1","chain":null}},{"id":"l-c04eabd4080c0076","kind":"oeis_match","node_a_id":"n-1e7342ec7a250847","node_b_id":null,"fingerprint":null,"oeis_id":"A084601","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":"A084601","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A084601","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-1e7342ec7a250847","oeis_id":"A084601","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":27,"oeis_terms_available":27,"term_count":200,"raw_size":12338}],"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;\n  2:    3,    2;\n  3:    7,    6;\n  4:   19,   24,    6;\n  5:   51,   80,   30;\n  6:  141,  270,  150,   20;\n  7:  393,  882,  630,  140;\n  8: 1107, 2856, 2520,  840,  70;\n  9: 3139, 9144, 9576, 4200, 630;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (5*n-4+(4*n-4)*x)*P(n-1,x) - (6+14*x+8*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n - 4 + (4*n - 4)*x)*P[n - 1, x] - (6 + 14*x + 8*x^2)*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((5*n - 4 + (4*n - 4)*x)*v[i - 1] - (6 + 14*x + 8*x^2)*deriv(v[i - 1], x))/(n)); 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-75ccd0453a9b6cec","steps":["\u2299 C(n,k)","x \u21a6 x+1"],"edges":["e-2126fae20f2e5116","e-0f1b9062159c7e91"],"text":"A011973 \u2192 \u2299 C(n,k) \u2192 x \u21a6 x+1"},"headline":"A011973 \u2192 \u2299 C(n,k) \u2192 x \u21a6 x+1","chain":null}
