{"id":"n-c1e8f5a3e0c5db28","revision":0,"first_index":0,"fingerprint":"f6-24cf4d809f9b6fe7","name":"shift_x_plus_one of gamma_polynomial of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","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"],["18","12"],["140","120"],["1330","1680","420"],["12852","20160","7560"],["130284","249480","138600","18480"],["1348776","3027024","2162160","480480"],["14247090","36756720","32432400","10810800","900900"],["152618180","444581280","465585120","204204000","30630600"],["1654120468","5370856920","6518191680","3569485920","814773960","46558512"],["18096447096","64793929200","89392343040","58663725120","17925027120","1955457504"],["199536967084","781003519296","1207378612440","924496853280","356056220520","59967363456","2498640144"],["2214714164600","9407217892800","16107305214000","14099469384000","6558930378000","1499184086400","124932007200"],["24720932068200","113249242033200","212750284546800","209618060652000","114446642310000","33249761344800","4336925392800","137680171200"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","30","260","3430","40572","536844","7018440","95147910","1297619180","17973987460","250826929080","3530938176220","50011753127000","712509528519000"],"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"],["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","8"],["-26","-24"],["20","16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["12","28","16"],["-24","-56","-32"]]},"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-86c00447f74e63c7","recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"]],"recurrence":{"denominator":{"coeffs":[["0"],["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["36","48"],["-96","-128"],["48","64"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2"],["-8"],["8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184068,"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-f4ffd13ebd20f033"]},{"id":184069,"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-e3db3509887403df","e-f4ffd13ebd20f033"]},{"id":184070,"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-e3db3509887403df","e-f4ffd13ebd20f033"]},{"id":184071,"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-e3db3509887403df"]},{"id":184072,"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-e3db3509887403df"]}],"region_exact":{},"incoming_edges":[{"id":"e-e3db3509887403df","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-c1e8f5a3e0c5db28","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":"5ec95d292bd2365507efeda2803b1fefade11e85e0c327ba5a80e3f4e6aebb6a","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-ce7f61ca1eb91390","role":"sequence","name":"gamma_polynomial of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","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-86c00447f74e63c7","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-c1e8f5a3e0c5db28","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":"2dd4eb337db2e4e1804bd107a345f29f94786918d6e81813f859edaa0007c115","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-f4ffd13ebd20f033","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-c1e8f5a3e0c5db28","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":"e6edc1e0c7f12b55d15b03beaa0309e10d81f6d5c7cc4c887879b0f36a229958","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-ce7f61ca1eb91390","role":"sequence","name":"gamma_polynomial of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","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-be3ec677cd2b09f2","kind":"fingerprint_related","node_a_id":"n-123dac0e8ec7cfb1","node_b_id":"n-c1e8f5a3e0c5db28","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]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4b168caf0bc63f8b","steps":["x \u21a6 x+1"],"edges":["e-502ba5da9494e9e3"],"text":"A089627 \u2192 x \u21a6 x+1"},"headline":"A089627 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-c2bd9581193792aa","kind":"fingerprint_related","node_a_id":"n-fc6d8de3e658fad0","node_b_id":"n-c1e8f5a3e0c5db28","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-fc6d8de3e658fad0","other":{"id":"n-fc6d8de3e658fad0","name":"mobius_x_over_x_plus_one of Triangle read by rows, T(n,k) = C(n,k)*C(k,n-k).","first_index":0,"fingerprint":"f6-24cf4d809f9b6fe7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","2","3"]],"card_rows":[["1"],["0","1"],["0","2","3"],["0","0","6","7"]],"row_sums_preview":["1","1","5","13","49","161","581","2045"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6fa6977267f2e73d","steps":["x \u21a6 x/(x+1)"],"edges":["e-804c4ca4aefa439b"],"text":"A105868 \u2192 x \u21a6 x/(x+1)"},"headline":"A105868 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-f6dfe6ec12109b7a","kind":"fingerprint_related","node_a_id":"n-1e7342ec7a250847","node_b_id":"n-c1e8f5a3e0c5db28","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-1e7342ec7a250847","other":{"id":"n-1e7342ec7a250847","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).","first_index":0,"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]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8bfdd58279be42d7","steps":["x \u21a6 x+1"],"edges":["e-0f1b9062159c7e91"],"text":"A391403 \u2192 x \u21a6 x+1"},"headline":"A391403 \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:         1;\n  1:         2;\n  2:        18,        12;\n  3:       140,       120;\n  4:      1330,      1680,       420;\n  5:     12852,     20160,      7560;\n  6:    130284,    249480,    138600,     18480;\n  7:   1348776,   3027024,   2162160,    480480;\n  8:  14247090,  36756720,  32432400,  10810800,   900900;\n  9: 152618180, 444581280, 465585120, 204204000, 30630600;\n","recurrence_sentence":"The row polynomials satisfy n^2*P(n,x) = (20*n^2-26*n+8+(16*n^2-24*n+8)*x)*P(n-1,x) - (24*n-12+(56*n-28)*x+(32*n-16)*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[((20*n^2 - 26*n + 8 + (16*n^2 - 24*n + 8)*x)*P[n - 1, x] - (24*n - 12 + (56*n - 28)*x + (32*n - 16)*x^2)*D[P[n - 1, x], x])/(n^2)]];\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] = ((20*n^2 - 26*n + 8 + (16*n^2 - 24*n + 8)*x)*v[i - 1] - (24*n - 12 + (56*n - 28)*x + (32*n - 16)*x^2)*deriv(v[i - 1], x))/(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-fdc03fad975441d0","steps":["\u03b3","basis A007318"],"edges":["e-510754e51c3ada00","e-e3db3509887403df"],"text":"A069466 \u2192 \u03b3 \u2192 basis A007318"},"headline":"A069466 \u2192 \u03b3 \u2192 basis A007318","chain":null}
