{"id":"n-255ef16c4f62b1af","revision":0,"first_index":0,"fingerprint":"f6-3ac782335c346aee","name":"basis_map_A013609 of reversal of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","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":[["8"],["30","24"],["112","192","64"],["420","1120","840","160"],["1584","5760","6912","3072","384"],["6006","27720","46200","33600","10080","896"],["22880","128128","274560","281600","140800","30720","2048"],["87516","576576","1513512","2018016","1441440","532224","88704","4608"],["335920","2545920","7920640","13045760","12230400","6522880","1863680","245760","10240"],["1293292","11085360","39907296","78414336","91483392","64576512","26906880","6150144","658944","22528"],["4992288","47752320","195350400","446515200","625121280","552738816","307077120","103219200","19353600","1720320","49152"],["19315400","203970624","934865360","2438779200","3990729600","4256778240","2979744768","1344245760","373401600","58572800","4392960","106496"],["74884320","865329920","4393213440","12886759424","24162673920","30255869952","25671647232","14669512704","5501067264","1286799360","171573248","11010048","229376"],["290845350","3650610600","20339116200","66290452800","140229804000","201930917760","201930917760","140473681920","67044257280","21283891200","4256778240","488816640","27156480","491520"],["1131445440","15329260800","92997515520","333508331520","786126781440","1281095495680","1478187110400","1216336822272","709529812992","287925141504","78525038592","13597409280","1359740928","66060288","1048576"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["8","54","368","2540","17712","124502","880736","6262596","44721200","320498684","2303889696","16604902808","119950570208","868237737750","6295717015232"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["8"],["30","24"]],"recurrence":{"denominator":{"coeffs":[["0"],["3"],["10"],["9"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0","24"],["0","0","28"],["0","0","-16"],["0","0","-28"],["0","0","-8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["12","0"],["38","32"],["32","32"],["8","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199644,"position":0,"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-ec102c83c39c9ffe"]},{"id":199645,"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-ec102c83c39c9ffe"]},{"id":199646,"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-ec102c83c39c9ffe"]},{"id":199647,"position":3,"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-ec102c83c39c9ffe"]}],"region_exact":{},"incoming_edges":[{"id":"e-ec102c83c39c9ffe","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-255ef16c4f62b1af","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":"c0b23930863c59051d78ad25b2e168d9281e22b961651d1705221b9f434148f9","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-4d976aa636e25300","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","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-1d23245308437939","kind":"fingerprint_related","node_a_id":"n-780354e043925a7c","node_b_id":"n-255ef16c4f62b1af","fingerprint":"f6-3ac782335c346aee","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3ac782335c346aee"},"other_id":"n-780354e043925a7c","other":{"id":"n-780354e043925a7c","name":"derivative of basis_map_A013609 of Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","first_index":0,"fingerprint":"f6-3ac782335c346aee","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["10","8"]],"card_rows":[["2"],["10","8"],["42","72","24"],["168","448","336","64"]],"row_sums_preview":["0","2","18","138","1016","7380","53358","385322"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c2eaf5212a4dec13","steps":["basis A013609","D"],"edges":["e-1b3fae304d93e9a3","e-eb8eb6667e3750b7"],"text":"A001263 \u2192 basis A013609 \u2192 D"},"headline":"A001263 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-1dd0a43d906b496a","kind":"fingerprint_related","node_a_id":"n-6decca65eb6ed468","node_b_id":"n-255ef16c4f62b1af","fingerprint":"f6-3ac782335c346aee","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3ac782335c346aee"},"other_id":"n-6decca65eb6ed468","other":{"id":"n-6decca65eb6ed468","name":"basis_map_A013609 of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-3ac782335c346aee","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["5","4"],["14","24","8"]],"card_rows":[["2"],["5","4"],["14","24","8"],["42","112","84","16"]],"row_sums_preview":["2","9","46","254","1476","8893","55046","347922"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["basis A013609"],"edges":["e-065d9f9c4e446794"],"text":"A108838 \u2192 basis A013609"},"headline":"A108838 \u2192 basis A013609","chain":null}},{"id":"l-efe9ab27a5324b08","kind":"fingerprint_related","node_a_id":"n-25810b399b931dcd","node_b_id":"n-255ef16c4f62b1af","fingerprint":"f6-3ac782335c346aee","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3ac782335c346aee"},"other_id":"n-25810b399b931dcd","other":{"id":"n-25810b399b931dcd","name":"derivative of basis_map_A013609 of narayana","first_index":1,"fingerprint":"f6-3ac782335c346aee","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["10","8"]],"card_rows":[["2"],["10","8"],["42","72","24"],["168","448","336","64"]],"row_sums_preview":["0","2","18","138","1016","7380","53358","385322"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["basis A013609","D"],"edges":["e-2f325d6b3b1beb16","e-d83149429d440c34"],"text":"narayana \u2192 basis A013609 \u2192 D"},"headline":"narayana \u2192 basis A013609 \u2192 D","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:       8;\n  1:      30,       24;\n  2:     112,      192,       64;\n  3:     420,     1120,      840,      160;\n  4:    1584,     5760,     6912,     3072,      384;\n  5:    6006,    27720,    46200,    33600,    10080,      896;\n  6:   22880,   128128,   274560,   281600,   140800,    30720,     2048;\n  7:   87516,   576576,  1513512,  2018016,  1441440,   532224,    88704,    4608;\n  8:  335920,  2545920,  7920640, 13045760, 12230400,  6522880,  1863680,  245760,  10240;\n  9: 1293292, 11085360, 39907296, 78414336, 91483392, 64576512, 26906880, 6150144, 658944, 22528;\n","recurrence_sentence":"The row polynomials satisfy (2*n^4+9*n^3+10*n^2+3*n)*P(n,x) = -(8*n^4+28*n^3+16*n^2-28*n-24)*x^2*P(n-2,x) + (8*n^4+32*n^3+38*n^2+12*n+(8*n^4+32*n^3+32*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 8, P(1,x) = 30 + 24*x.","mathematica":"P[0, x_] := 8;\nP[1, x_] := 30 + 24*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^4 + 28*n^3 + 16*n^2 - 28*n - 24)*x^2*P[n - 2, x] + (8*n^4 + 32*n^3 + 38*n^2 + 12*n + (8*n^4 + 32*n^3 + 32*n^2)*x)*P[n - 1, x])/(2*n^4 + 9*n^3 + 10*n^2 + 3*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 8; v[2] = 30 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^4 + 28*n^3 + 16*n^2 - 28*n - 24)*x^2*v[i - 2] + (8*n^4 + 32*n^3 + 38*n^2 + 12*n + (8*n^4 + 32*n^3 + 32*n^2)*x)*v[i - 1])/(2*n^4 + 9*n^3 + 10*n^2 + 3*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-7a4827a398e28fe4","steps":["reverse","basis A013609"],"edges":["e-8b10d4d15ed56acc","e-ec102c83c39c9ffe"],"text":"A352880 \u2192 reverse \u2192 basis A013609"},"headline":"A352880 \u2192 reverse \u2192 basis A013609","chain":null}
