{"id":"n-a60016fa379da581","revision":0,"first_index":0,"fingerprint":"f6-2c1fea4a558d4e6c","name":"shift_x_plus_one of theta_plus_one 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.","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"],["3","2"],["10","12","3"],["35","60","30","4"],["126","280","210","60","5"],["462","1260","1260","560","105","6"],["1716","5544","6930","4200","1260","168","7"],["6435","24024","36036","27720","11550","2520","252","8"],["24310","102960","180180","168168","90090","27720","4620","360","9"],["92378","437580","875160","960960","630630","252252","60060","7920","495","10"],["352716","1847560","4157010","5250960","4084080","2018016","630630","120120","12870","660","11"],["1352078","7759752","19399380","27713400","24942060","14702688","5717712","1441440","225225","20020","858","12"],["5200300","32449872","89237148","142262120","145495350","99768240","46558512","14702688","3063060","400400","30030","1092","13"],["20058300","135207800","405623400","713897184","818007190","640179540","349188840","133024320","34918884","6126120","680680","43680","1365","14"],["77558760","561632400","1825305300","3515402800","4461857400","3926434512","2454021570","1097450640","349188840","77597520","11639628","1113840","61880","1680","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","25","129","681","3653","19825","108545","598417","3317445","18474633","103274625","579168825","3256957317","18359266785"],"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"],["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["2","1"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-1"],["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175019,"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-09282c699b4099b7"]},{"id":175020,"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-09282c699b4099b7"]},{"id":175021,"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-09282c699b4099b7"]},{"id":175022,"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-09282c699b4099b7"]}],"region_exact":{},"incoming_edges":[{"id":"e-09282c699b4099b7","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-a60016fa379da581","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":"497c40c2b0e3289e9e5205e2e1f6f32791cbfab3998f639517bc120af6da6aef","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-16e42f812cdd0509","role":"sequence","name":"theta_plus_one 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.","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-17d8438f9479f7ab","kind":"fingerprint_related","node_a_id":"n-a60016fa379da581","node_b_id":"n-133f3a16dadf73ac","fingerprint":"f6-2c1fea4a558d4e6c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2c1fea4a558d4e6c"},"other_id":"n-133f3a16dadf73ac","other":{"id":"n-133f3a16dadf73ac","name":"reversal of Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","first_index":1,"fingerprint":"f6-2c1fea4a558d4e6c","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["3","2"]],"card_rows":[["1"],["1"],["3","2"],["10","12","3"]],"row_sums_preview":["1","1","5","25","129","681","3653","19825"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b0d6ec97a8d172b8","steps":["reverse"],"edges":["e-70add7cce9667674"],"text":"A253283 \u2192 reverse"},"headline":"A253283 \u2192 reverse","chain":null}},{"id":"l-5b51f1f122652cb5","kind":"fingerprint_related","node_a_id":"n-a60016fa379da581","node_b_id":"n-43c5f46a251ccde6","fingerprint":"f6-2c1fea4a558d4e6c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2c1fea4a558d4e6c"},"other_id":"n-43c5f46a251ccde6","other":{"id":"n-43c5f46a251ccde6","name":"reversal of Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.","first_index":0,"fingerprint":"f6-2c1fea4a558d4e6c","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","3"]],"card_rows":[["1"],["1"],["2","3"],["3","12","10"]],"row_sums_preview":["1","1","5","25","129","681","3653","19825"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0a5e54eacdb24124","steps":["reverse"],"edges":["e-a6f5a3bfa125593b"],"text":"A360282 \u2192 reverse"},"headline":"A360282 \u2192 reverse","chain":null}},{"id":"l-62790b8a0009afd7","kind":"fingerprint_related","node_a_id":"n-c2b0d68b9e568dde","node_b_id":"n-a60016fa379da581","fingerprint":"f6-2c1fea4a558d4e6c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2c1fea4a558d4e6c"},"other_id":"n-c2b0d68b9e568dde","other":{"id":"n-c2b0d68b9e568dde","name":"basis_map_A007318 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","first_index":0,"fingerprint":"f6-2c1fea4a558d4e6c","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["3","2"]],"card_rows":[["1"],["1"],["3","2"],["10","12","3"]],"row_sums_preview":["1","1","5","25","129","681","3653","19825"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b1b19325e0094118","steps":["x \u21a6 x+1"],"edges":["e-0de802f77814f187"],"text":"A367270 \u2192 x \u21a6 x+1"},"headline":"A367270 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-82795db91991d90f","kind":"fingerprint_related","node_a_id":"n-a60016fa379da581","node_b_id":"n-4cab0236cf9d2d4f","fingerprint":"f6-2c1fea4a558d4e6c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2c1fea4a558d4e6c"},"other_id":"n-4cab0236cf9d2d4f","other":{"id":"n-4cab0236cf9d2d4f","name":"derivative of Triangle read by rows: T(n,k) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.","first_index":1,"fingerprint":"f6-2c1fea4a558d4e6c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["4","6"]],"card_rows":[["1"],["4","6"],["9","36","30"],["16","120","240","140"]],"row_sums_preview":["0","1","10","75","516","3405","21918","138775"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-55697ddae4d22d60","steps":["D"],"edges":["e-5b009bca5501b390"],"text":"A123160 \u2192 D"},"headline":"A123160 \u2192 D","chain":null}},{"id":"l-b52c990827a8dfd3","kind":"fingerprint_related","node_a_id":"n-b0d6ec97a8d172b8","node_b_id":"n-a60016fa379da581","fingerprint":"f6-2c1fea4a558d4e6c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2c1fea4a558d4e6c"},"other_id":"n-b0d6ec97a8d172b8","other":{"id":"n-b0d6ec97a8d172b8","name":"Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","first_index":1,"fingerprint":"f6-2c1fea4a558d4e6c","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","3","12","10"]],"row_sums_preview":["1","1","5","25","129","681","3653","19825"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A253283"],"derivation":{"root":"n-b0d6ec97a8d172b8","steps":[],"edges":[],"text":"A253283"},"headline":"A253283: Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","chain":null}},{"id":"l-d96372df48b9db2a","kind":"fingerprint_related","node_a_id":"n-0a5e54eacdb24124","node_b_id":"n-a60016fa379da581","fingerprint":"f6-2c1fea4a558d4e6c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2c1fea4a558d4e6c"},"other_id":"n-0a5e54eacdb24124","other":{"id":"n-0a5e54eacdb24124","name":"Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.","first_index":0,"fingerprint":"f6-2c1fea4a558d4e6c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","3","2"]],"card_rows":[["1"],["0","1"],["0","3","2"],["0","10","12","3"]],"row_sums_preview":["1","1","5","25","129","681","3653","19825"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A360282"],"derivation":{"root":"n-0a5e54eacdb24124","steps":[],"edges":[],"text":"A360282"},"headline":"A360282: Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 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:     3,      2;\n  2:    10,     12,      3;\n  3:    35,     60,     30,      4;\n  4:   126,    280,    210,     60,      5;\n  5:   462,   1260,   1260,    560,    105,      6;\n  6:  1716,   5544,   6930,   4200,   1260,    168,     7;\n  7:  6435,  24024,  36036,  27720,  11550,   2520,   252,    8;\n  8: 24310, 102960, 180180, 168168,  90090,  27720,  4620,  360,   9;\n  9: 92378, 437580, 875160, 960960, 630630, 252252, 60060, 7920, 495, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n)*P(n,x) = (4*n^2+2*n+(3*n^2+n)*x)*P(n-1,x) - ((2*n+1)*x+(2*n+1)*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[((4*n^2 + 2*n + (3*n^2 + n)*x)*P[n - 1, x] - ((2*n + 1)*x + (2*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 + n)]];\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] = ((4*n^2 + 2*n + (3*n^2 + n)*x)*v[i - 1] - ((2*n + 1)*x + (2*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 + 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-c2eaf5212a4dec13","steps":["\u03b8+1","x \u21a6 x+1"],"edges":["e-127d139330d3e4d4","e-09282c699b4099b7"],"text":"A001263 \u2192 \u03b8+1 \u2192 x \u21a6 x+1"},"headline":"A001263 \u2192 \u03b8+1 \u2192 x \u21a6 x+1","chain":null}
