{"id":"n-50746bd627deaa2a","revision":0,"first_index":0,"fingerprint":"f6-65f5fe102fc0d40e","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","1"],["6","6","1"],["14","21","9","1"],["34","68","46","12","1"],["82","205","190","80","15","1"],["198","594","699","408","123","18","1"],["478","1673","2387","1785","749","175","21","1"],["1154","4616","7740","7064","3814","1240","236","24","1"],["2786","12537","24156","26040","17226","7218","1908","306","27","1"],["6726","33630","73205","91040","71370","36716","12510","2780","385","30","1"],["16238","89309","216733","305481","277046","169246","70862","20284","3883","473","33","1"],["39202","235212","629610","991940","1021983","723624","360196","126720","31215","5244","570","36","1"],["94642","615173","1800474","3135704","3618433","2915523","1684124","704782","213317","46059","6890","676","39","1"],["228486","1599402","5080943","9693432","12386493","11195086","7367591","3580604","1289351","341894","65653","8848","791","42","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","13","45","161","573","2041","7269","25889","92205","328393","1169589","4165553","14835837","52838617"],"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","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["3"],["-5"],["2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["20","21","3"],["-8","-8","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["8","16","9","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-652a7cfa737886da","recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"],["6","6","1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-2"],["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","-2"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172669,"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-d144e18e9c7a81c4"]},{"id":172670,"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-9141b466b4e3524d","e-d144e18e9c7a81c4"]},{"id":172671,"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-9141b466b4e3524d","e-d144e18e9c7a81c4"]},{"id":172672,"position":3,"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-9141b466b4e3524d","e-d144e18e9c7a81c4"]},{"id":172673,"position":4,"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-9141b466b4e3524d"]}],"region_exact":{},"incoming_edges":[{"id":"e-9141b466b4e3524d","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-50746bd627deaa2a","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":"8e6dad8922d9ff7964c743ac1bbd63d8226bcb5a3b5f8a1741624ad0f2f76d0a","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-9d2e609ab19d361f","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= 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-652a7cfa737886da","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-50746bd627deaa2a","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":"31eace9f8b5c812928fe6379c86907fb88303f8630c83ca041f68e34a066076f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-d144e18e9c7a81c4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-50746bd627deaa2a","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":"ef86665f0f2823a56b81046b0cdcd9896f75fb90efb03a73b072f033d8226f12","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9d2e609ab19d361f","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= 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-3be41adcac98e833","kind":"fingerprint_related","node_a_id":"n-50746bd627deaa2a","node_b_id":"n-9a90d366df7da363","fingerprint":"f6-65f5fe102fc0d40e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-65f5fe102fc0d40e"},"other_id":"n-9a90d366df7da363","other":{"id":"n-9a90d366df7da363","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).","first_index":0,"fingerprint":"f6-65f5fe102fc0d40e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","6"]],"card_rows":[["1"],["1","2"],["1","6","6"],["1","9","21","14"]],"row_sums_preview":["1","3","13","45","161","573","2041","7269"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9d2e609ab19d361f","steps":["x \u21a6 x/(x+1)"],"edges":["e-4960bec3ed0d61d9"],"text":"A102413 \u2192 x \u21a6 x/(x+1)"},"headline":"A102413 \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,     1;\n  2:    6,     6,     1;\n  3:   14,    21,     9,     1;\n  4:   34,    68,    46,    12,     1;\n  5:   82,   205,   190,    80,    15,    1;\n  6:  198,   594,   699,   408,   123,   18,    1;\n  7:  478,  1673,  2387,  1785,   749,  175,   21,   1;\n  8: 1154,  4616,  7740,  7064,  3814, 1240,  236,  24,  1;\n  9: 2786, 12537, 24156, 26040, 17226, 7218, 1908, 306, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n^2-5*n+3)*P(n-1,x) - (8*n-20+(8*n-21)*x+(n-3)*x^2)*d/dx P(n-1,x) + (8+16*x+9*x^2+x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 3)*P[n - 1, x] - (8*n - 20 + (8*n - 21)*x + (n - 3)*x^2)*D[P[n - 1, x], x] + (8 + 16*x + 9*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 - 5*n + 3)*v[i - 1] - (8*n - 20 + (8*n - 21)*x + (n - 3)*x^2)*deriv(v[i - 1], x) + (8 + 16*x + 9*x^2 + x^3)*deriv(deriv(v[i - 1], x), 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-9d2e609ab19d361f","steps":["basis A007318"],"edges":["e-9141b466b4e3524d"],"text":"A102413 \u2192 basis A007318"},"headline":"A102413 \u2192 basis A007318","chain":null}