{"id":"n-9cc1c067e5b017e3","revision":0,"first_index":0,"fingerprint":"f6-ade972e3543234ee","name":"reversal of theta_plus_one of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-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"],["4","1"],["12","12","1"],["32","72","24","1"],["80","320","240","40","1"],["192","1200","1600","600","60","1"],["448","4032","8400","5600","1260","84","1"],["1024","12544","37632","39200","15680","2352","112","1"],["2304","36864","150528","225792","141120","37632","4032","144","1"],["5120","103680","552960","1128960","1016064","423360","80640","6480","180","1"],["11264","281600","1900800","5068800","6209280","3725568","1108800","158400","9900","220","1"],["24576","743424","6195200","20908800","33454080","27320832","11708928","2613600","290400","14520","264","1"],["53248","1916928","19329024","80537600","163088640","173961216","101477376","32617728","5662800","503360","20592","312","1"],["114688","4845568","58146816","293156864","732892160","989404416","753831936","329801472","82450368","11451440","832832","28392","364","1"],["245760","12042240","169594880","1017569280","3078147072","5130245120","4947022080","2826869760","961920960","192384192","21861840","1324960","38220","420","1"]],"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:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","-1"],["4","0"],["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-1"],["0","4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178683,"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-580d3e1c63295286"]},{"id":178684,"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-580d3e1c63295286"]},{"id":178685,"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-580d3e1c63295286"]}],"region_exact":{},"incoming_edges":[{"id":"e-580d3e1c63295286","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-9cc1c067e5b017e3","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"e2db75496a619bedb515125023235ce4ca7dee648a4e832f45c3be19ef5076ba","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-916d0051af2e0118","role":"sequence","name":"theta_plus_one of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-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-6851d7c839fdd368","kind":"fingerprint_related","node_a_id":"n-587271daecb2c809","node_b_id":"n-9cc1c067e5b017e3","fingerprint":"f6-ade972e3543234ee","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ade972e3543234ee"},"other_id":"n-587271daecb2c809","other":{"id":"n-587271daecb2c809","name":"theta_plus_one of reversal of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","first_index":1,"fingerprint":"f6-ade972e3543234ee","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","12","12"]],"card_rows":[["1"],["1","4"],["1","12","12"],["1","24","72","32"]],"row_sums_preview":["1","5","25","129","681","3653","19825","108545"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0d75f6272d2a2a84","steps":["reverse","\u03b8+1"],"edges":["e-87267df0bda73e27","e-bd31a37f32a8c0aa"],"text":"A114656 \u2192 reverse \u2192 \u03b8+1"},"headline":"A114656 \u2192 reverse \u2192 \u03b8+1","chain":null}},{"id":"l-b0cbdabe17418995","kind":"fingerprint_related","node_a_id":"n-d6831a2f321e9d14","node_b_id":"n-9cc1c067e5b017e3","fingerprint":"f6-ade972e3543234ee","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ade972e3543234ee"},"other_id":"n-d6831a2f321e9d14","other":{"id":"n-d6831a2f321e9d14","name":"theta_plus_one of reversal of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.","first_index":1,"fingerprint":"f6-ade972e3543234ee","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["2","8"],["2","24","24"]],"card_rows":[["2"],["2","8"],["2","24","24"],["2","48","144","64"]],"row_sums_preview":["2","10","50","258","1362","7306","39650","217090"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a30765140629bd5","steps":["reverse","\u03b8+1"],"edges":["e-a35c7072fcaf677b","e-101a9c96b8392e2e"],"text":"A114655 \u2192 reverse \u2192 \u03b8+1"},"headline":"A114655 \u2192 reverse \u2192 \u03b8+1","chain":null}},{"id":"l-f0989c10ab9e64ed","kind":"fingerprint_related","node_a_id":"n-916d0051af2e0118","node_b_id":"n-9cc1c067e5b017e3","fingerprint":"f6-ade972e3543234ee","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ade972e3543234ee"},"other_id":"n-916d0051af2e0118","other":{"id":"n-916d0051af2e0118","name":"theta_plus_one of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","first_index":0,"fingerprint":"f6-ade972e3543234ee","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","12","12"]],"card_rows":[["1"],["1","4"],["1","12","12"],["1","24","72","32"]],"row_sums_preview":["1","5","25","129","681","3653","19825","108545"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8ce61d302cec295b","steps":["\u03b8+1"],"edges":["e-2a32d110f0351649"],"text":"A114687 \u2192 \u03b8+1"},"headline":"A114687 \u2192 \u03b8+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:    4,      1;\n  2:   12,     12,      1;\n  3:   32,     72,     24,       1;\n  4:   80,    320,    240,      40,       1;\n  5:  192,   1200,   1600,     600,      60,      1;\n  6:  448,   4032,   8400,    5600,    1260,     84,     1;\n  7: 1024,  12544,  37632,   39200,   15680,   2352,   112,    1;\n  8: 2304,  36864, 150528,  225792,  141120,  37632,  4032,  144,   1;\n  9: 5120, 103680, 552960, 1128960, 1016064, 423360, 80640, 6480, 180, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n)*P(n,x) = (2*n^2+4*n+2+(3*n^2-1)*x)*P(n-1,x) + ((4*n+2)*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[((2*n^2 + 4*n + 2 + (3*n^2 - 1)*x)*P[n - 1, x] + ((4*n + 2)*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] = ((2*n^2 + 4*n + 2 + (3*n^2 - 1)*x)*v[i - 1] + ((4*n + 2)*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-8ce61d302cec295b","steps":["\u03b8+1","reverse"],"edges":["e-2a32d110f0351649","e-580d3e1c63295286"],"text":"A114687 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A114687 \u2192 \u03b8+1 \u2192 reverse","chain":null}