{"id":"n-3ad802ff9e828000","revision":0,"first_index":0,"fingerprint":"f6-935c57c7e92571a4","name":"shift_x_plus_one of reversal of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["11","8","1"],["41","47","15","1"],["153","240","126","24","1"],["571","1137","834","270","35","1"],["2131","5144","4851","2232","505","48","1"],["7953","22559","26013","15555","5075","861","63","1"],["29681","96736","131900","97168","41590","10304","1372","80","1"],["110771","407841","641988","562332","298498","97830","19236","2076","99","1"],["413403","1696808","3028581","3075600","1950830","797664","209034","33648","3015","120","1"],["1542841","6984655","13939499","16105413","11889370","5835158","1918070","414282","55869","4235","143","1"],["5757961","28500688","62891706","81471080","68637695","39344448","15587404","4243888","772695","88880","5786","168","1"],["21489003","115448337","279116550","400698666","379453245","248933871","115560756","38064780","8776053","1370655","136422","7722","195","1"],["80198051","464754744","1221711831","1925272584","2024738751","1496551056","797196771","308617488","86385897","17156568","2330757","203112","10101","224","1"],["299303201","1860988735","5284854585","9070133895","10489533045","8627142699","5189007453","2310743331","762312195","184410525","31982379","3822741","294567","12985","255","1"],["1117014753","7417410496","22629976440","42016793760","53001404220","48016335744","32195277576","16221426144","6169917798","1763720640","373720776","57234528","6076668","417536","16440","288","1"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["20","104","544","2848","14912","78080","408832","2140672","11208704","58689536","307302400","1609056256","8425127936","44114542592","230986743808"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["11","8","1"]],"recurrence":{"denominator":{"coeffs":[["3"],["2"]]},"terms":[{"coeff":{"coeffs":[["13","7"],["10","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","-10","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177618,"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-39a2dcd907acf65d"]},{"id":177619,"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-39a2dcd907acf65d"]},{"id":177620,"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-39a2dcd907acf65d"]}],"region_exact":{},"incoming_edges":[{"id":"e-39a2dcd907acf65d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-3ad802ff9e828000","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":"6d8fd95dc1995672775c2fe9d2907f395e0d4d76b821cf7ff9b77e0c160e1568","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-ccea8be4a0f670ab","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (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-a1eb17914324fd33","kind":"fingerprint_related","node_a_id":"n-3ad802ff9e828000","node_b_id":"n-2b6c5dd961f03a94","fingerprint":"f6-935c57c7e92571a4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-935c57c7e92571a4"},"other_id":"n-2b6c5dd961f03a94","other":{"id":"n-2b6c5dd961f03a94","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","first_index":0,"fingerprint":"f6-935c57c7e92571a4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","2"],["44","32","4"]],"card_rows":[["1"],["6","2"],["44","32","4"],["328","376","120","8"]],"row_sums_preview":["1","8","80","832","8704","91136","954368","9994240"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c30cb06ad75c5e21","steps":["basis A013609","basis A007318"],"edges":["e-49766d0fffe14f49","e-5d45ef911b70a650"],"text":"A103327 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A103327 \u2192 basis A013609 \u2192 basis A007318","chain":null}},{"id":"l-e6f2c4890e2965ae","kind":"fingerprint_related","node_a_id":"n-f356d4c20400a3cf","node_b_id":"n-3ad802ff9e828000","fingerprint":"f6-935c57c7e92571a4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-935c57c7e92571a4"},"other_id":"n-f356d4c20400a3cf","other":{"id":"n-f356d4c20400a3cf","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","first_index":0,"fingerprint":"f6-935c57c7e92571a4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1","8","11"],["1","15","47","41"],["1","24","126","240","153"]],"card_rows":[["1","8","11"],["1","15","47","41"],["1","24","126","240","153"],["1","35","270","834","1137","571"]],"row_sums_preview":["20","104","544","2848","14912","78080","408832","2140672"],"regions":["[-1, 0)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8eb9bb3a09f0b7fd","steps":["x \u21a6 x/(x+1)"],"edges":["e-10dce9fcf56455e3"],"text":"A123519 \u2192 x \u21a6 x/(x+1)"},"headline":"A123519 \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:      11,       8,        1;\n  1:      41,      47,       15,        1;\n  2:     153,     240,      126,       24,        1;\n  3:     571,    1137,      834,      270,       35,       1;\n  4:    2131,    5144,     4851,     2232,      505,      48,       1;\n  5:    7953,   22559,    26013,    15555,     5075,     861,      63,      1;\n  6:   29681,   96736,   131900,    97168,    41590,   10304,    1372,     80,     1;\n  7:  110771,  407841,   641988,   562332,   298498,   97830,   19236,   2076,    99,    1;\n  8:  413403, 1696808,  3028581,  3075600,  1950830,  797664,  209034,  33648,  3015,  120,   1;\n  9: 1542841, 6984655, 13939499, 16105413, 11889370, 5835158, 1918070, 414282, 55869, 4235, 143, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n+3)*P(n,x) = (10*n+13+(6*n+7)*x)*P(n-1,x) - (6+10*x+4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 8*x + x^2.","mathematica":"P[0, x_] := 11 + 8*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((10*n + 13 + (6*n + 7)*x)*P[n - 1, x] - (6 + 10*x + 4*x^2)*D[P[n - 1, x], x])/(2*n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 11 + 8*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((10*n + 13 + (6*n + 7)*x)*v[i - 1] - (6 + 10*x + 4*x^2)*deriv(v[i - 1], x))/(2*n + 3)); 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-8eb9bb3a09f0b7fd","steps":["reverse","x \u21a6 x+1"],"edges":["e-efd270083a78762c","e-39a2dcd907acf65d"],"text":"A123519 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A123519 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}
