{"id":"n-b2e66fb8e43e7448","revision":1,"first_index":0,"fingerprint":"f6-1e5fbd8fe1b88630","name":"mobius_x_over_x_plus_one of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","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","4"],["3","15","15"],["4","36","84","56"],["5","70","280","420","210"],["6","120","720","1800","1980","792"],["7","189","1575","5775","10395","9009","3003"],["8","280","3080","15400","40040","56056","40040","11440"],["9","396","5544","36036","126126","252252","288288","175032","43758"],["10","540","9360","76440","343980","917280","1485120","1432080","755820","167960"],["11","715","15015","150150","840840","2858856","6126120","8314020","6928350","3233230","646646"],["12","924","23100","277200","1884960","7916832","21488544","38372400","44767800","32829720","13728792","2496144"],["13","1170","34320","486200","3938220","19953648","66512160","149652360","228635550","233716340","152977968","57946200","9657700"],["14","1456","49504","816816","7759752","46558512","186234048","512143632","981608628","1308811504","1189828640","703080560","243374040","37442160"],["15","1785","69615","1322685","14549535","101846745","480134655","1577585295","3681032355","6135053925","7250518275","5932242225","3194284275","1017958725","145422675"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","33","180","985","5418","29953","166344","927441","5188590","29113953","163786428","923511849","5217709266","29532022785"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","4"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["2","4","2"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["-1","-2","-1"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","4","5","2"],["3","12","15","6"],["2","8","10","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164704,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-6a8f8db3c1596aaf"]},{"id":164705,"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-6a8f8db3c1596aaf"]},{"id":164706,"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-6a8f8db3c1596aaf"]}],"region_exact":{},"incoming_edges":[{"id":"e-6a8f8db3c1596aaf","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-b2e66fb8e43e7448","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"8f1d30d94507e9afdff6cdfa5398c8b7bc52fba8250cee4bd037daa8c8bed4d6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-984320118b653f6b","role":"sequence","name":"Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","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-4bb9438f72838d34","kind":"fingerprint_related","node_a_id":"n-6f30da2298fd30ba","node_b_id":"n-b2e66fb8e43e7448","fingerprint":"f6-1e5fbd8fe1b88630","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1e5fbd8fe1b88630"},"other_id":"n-6f30da2298fd30ba","other":{"id":"n-6f30da2298fd30ba","name":"basis_map_A007318 of narayana","first_index":1,"fingerprint":"f6-1e5fbd8fe1b88630","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["5","5","1"]],"card_rows":[["1"],["2","1"],["5","5","1"],["14","21","9","1"]],"row_sums_preview":["1","3","11","45","197","903","4279","20793"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["basis A007318"],"edges":["e-58b8cdf8af6ac600"],"text":"narayana \u2192 basis A007318"},"headline":"narayana \u2192 basis A007318","chain":null}},{"id":"l-834f3435b3e0725a","kind":"fingerprint_related","node_a_id":"n-b2e66fb8e43e7448","node_b_id":"n-68cd61e91317cdaa","fingerprint":"f6-1e5fbd8fe1b88630","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1e5fbd8fe1b88630"},"other_id":"n-68cd61e91317cdaa","other":{"id":"n-68cd61e91317cdaa","name":"derivative of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","first_index":1,"fingerprint":"f6-1e5fbd8fe1b88630","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["6","12"]],"card_rows":[["2"],["6","12"],["12","60","60"],["20","180","420","280"]],"row_sums_preview":["0","2","18","132","900","5910","37926","239624"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4689fb1386d440ad","steps":["D"],"edges":["e-ab4d580ed9bca928"],"text":"A063007 \u2192 D"},"headline":"A063007 \u2192 D","chain":null}},{"id":"l-ab1016e36c493802","kind":"fingerprint_related","node_a_id":"n-00a555c278779f26","node_b_id":"n-b2e66fb8e43e7448","fingerprint":"f6-1e5fbd8fe1b88630","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1e5fbd8fe1b88630"},"other_id":"n-00a555c278779f26","other":{"id":"n-00a555c278779f26","name":"shift_x_plus_one of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","first_index":0,"fingerprint":"f6-1e5fbd8fe1b88630","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["15","15","3"]],"card_rows":[["1"],["4","2"],["15","15","3"],["56","84","36","4"]],"row_sums_preview":["1","6","33","180","985","5418","29953","166344"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-984320118b653f6b","steps":["x \u21a6 x+1"],"edges":["e-f4ea45816c9a82eb"],"text":"A132812 \u2192 x \u21a6 x+1"},"headline":"A132812 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-b09a2abe3c11f26f","kind":"fingerprint_related","node_a_id":"n-4d61144188f12208","node_b_id":"n-b2e66fb8e43e7448","fingerprint":"f6-1e5fbd8fe1b88630","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1e5fbd8fe1b88630"},"other_id":"n-4d61144188f12208","other":{"id":"n-4d61144188f12208","name":"mobius_x_over_x_plus_one of narayana","first_index":1,"fingerprint":"f6-1e5fbd8fe1b88630","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","5"]],"card_rows":[["1"],["1","2"],["1","5","5"],["1","9","21","14"]],"row_sums_preview":["1","3","11","45","197","903","4279","20793"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["x \u21a6 x/(x+1)"],"edges":["e-50c973bfb8d0ccf1"],"text":"narayana \u2192 x \u21a6 x/(x+1)"},"headline":"narayana \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-cc3f398e7f32ff38","kind":"fingerprint_related","node_a_id":"n-66e2c3448245b0b7","node_b_id":"n-b2e66fb8e43e7448","fingerprint":"f6-1e5fbd8fe1b88630","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1e5fbd8fe1b88630"},"other_id":"n-66e2c3448245b0b7","other":{"id":"n-66e2c3448245b0b7","name":"Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","first_index":0,"fingerprint":"f6-1e5fbd8fe1b88630","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","5"]],"card_rows":[["1"],["1","2"],["1","5","5"],["1","9","21","14"]],"row_sums_preview":["1","3","11","45","197","903","4279","20793"],"regions":["(-inf, +inf)","(-inf, 0)","(-1, 0)","[-1, 0]","(-inf, 0]"],"region_exact":{},"oeis_ids":["A033282"],"derivation":{"root":"n-66e2c3448245b0b7","steps":[],"edges":[],"text":"A033282"},"headline":"A033282: Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","chain":null}},{"id":"l-e8fdee51edea565c","kind":"fingerprint_related","node_a_id":"n-57b4b14604f4c0d7","node_b_id":"n-b2e66fb8e43e7448","fingerprint":"f6-1e5fbd8fe1b88630","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1e5fbd8fe1b88630"},"other_id":"n-57b4b14604f4c0d7","other":{"id":"n-57b4b14604f4c0d7","name":"Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","first_index":0,"fingerprint":"f6-1e5fbd8fe1b88630","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["5","5","1"]],"card_rows":[["1"],["2","1"],["5","5","1"],["14","21","9","1"]],"row_sums_preview":["1","3","11","45","197","903","4279","20793"],"regions":["(-inf, +inf)","(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A126216"],"derivation":{"root":"n-57b4b14604f4c0d7","steps":[],"edges":[],"text":"A126216"},"headline":"A126216: Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","chain":null}},{"id":"l-2ad43bd49436cf77","kind":"oeis_match","node_a_id":"n-b2e66fb8e43e7448","node_b_id":null,"fingerprint":null,"oeis_id":"A050151","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A050151","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A050151","relation":"match_row_sums"}],"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,   4;\n  2:  3,  15,   15;\n  3:  4,  36,   84,    56;\n  4:  5,  70,  280,   420,    210;\n  5:  6, 120,  720,  1800,   1980,    792;\n  6:  7, 189, 1575,  5775,  10395,   9009,    3003;\n  7:  8, 280, 3080, 15400,  40040,  56056,   40040,   11440;\n  8:  9, 396, 5544, 36036, 126126, 252252,  288288,  175032,  43758;\n  9: 10, 540, 9360, 76440, 343980, 917280, 1485120, 1432080, 755820, 167960;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n+(2*n^2+4*n)*x+(n^2+2*n)*x^2)*P(n,x) = -(n^2+n+(2*n^2+2*n)*x+(n^2+n)*x^2)*P(n-2,x) + (2*n^2+3*n+1+(8*n^2+12*n+4)*x+(10*n^2+15*n+5)*x^2+(4*n^2+6*n+2)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + n + (2*n^2 + 2*n)*x + (n^2 + n)*x^2)*P[n - 2, x] + (2*n^2 + 3*n + 1 + (8*n^2 + 12*n + 4)*x + (10*n^2 + 15*n + 5)*x^2 + (4*n^2 + 6*n + 2)*x^3)*P[n - 1, x])/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + n + (2*n^2 + 2*n)*x + (n^2 + n)*x^2)*v[i - 2] + (2*n^2 + 3*n + 1 + (8*n^2 + 12*n + 4)*x + (10*n^2 + 15*n + 5)*x^2 + (4*n^2 + 6*n + 2)*x^3)*v[i - 1])/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*x^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-984320118b653f6b","steps":["x \u21a6 x/(x+1)"],"edges":["e-6a8f8db3c1596aaf"],"text":"A132812 \u2192 x \u21a6 x/(x+1)"},"headline":"A132812 \u2192 x \u21a6 x/(x+1)","chain":null}