{"id":"n-e09489fb7491a8fa","revision":0,"first_index":0,"fingerprint":"f6-48b123f09be537c0","name":"one_plus_derivative 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 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"],["3","1"],["10","8","1"],["32","48","15","1"],["96","240","140","24","1"],["272","1040","1000","320","35","1"],["736","4032","5880","3080","630","48","1"],["1920","14336","29792","23520","7840","1120","63","1"],["4864","47616","134400","150528","75264","17472","1848","80","1"],["12032","149760","552960","838656","592704","205632","35280","2880","99","1"],["29184","450560","2112000","4181760","3991680","1951488","498960","66000","4290","120","1"],["69632","1306624","7589120","19050240","23696640","15611904","5610528","1103520","116160","6160","143","1"],["163840","3674112","25918464","80537600","126846720","108725760","52550784","14496768","2265120","194480","8580","168","1"],["380928","10063872","84797440","319807488","622958336","675307776","424030464","157048320","34354320","4372368","312312","11648","195","1"],["876544","26951680","267438080","1204123648","2844954112","3811039232","3023180160","1452696960","425993568","75787712","8016008","484120","15470","224","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","19","96","501","2668","14407","78592","432073","2390004","13286043","74160672","415382397","2333445468","13141557519"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["8","-8","2"],["2","0","1"],["1","2","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["16","-16","4","0"],["12","4","-7","3"],["6","3","6","0"],["2","3","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-12","8","-2"],["0","2","7","-4","0"],["0","4","-2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176559,"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-86acb052ed946eeb"]},{"id":176560,"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-86acb052ed946eeb"]},{"id":176561,"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-86acb052ed946eeb"]},{"id":176562,"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-86acb052ed946eeb"]}],"region_exact":{},"incoming_edges":[{"id":"e-86acb052ed946eeb","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-e09489fb7491a8fa","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"d5a47b42bdecb6952414f492b7083d4aa488c78a898423c93d3f68b652e6bf26","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-1b0fc3e5a40ede1a","role":"sequence","name":"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 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-4a2a21a8ff91a5e2","kind":"fingerprint_related","node_a_id":"n-fb038b43f3455c33","node_b_id":"n-e09489fb7491a8fa","fingerprint":"f6-48b123f09be537c0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-48b123f09be537c0"},"other_id":"n-fb038b43f3455c33","other":{"id":"n-fb038b43f3455c33","name":"one_plus_derivative 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":"f6-48b123f09be537c0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["10","8","1"]],"card_rows":[["1"],["3","1"],["10","8","1"],["32","48","15","1"]],"row_sums_preview":["1","4","19","96","501","2668","14407","78592"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0d75f6272d2a2a84","steps":["D+1"],"edges":["e-094a3f846db3fe14"],"text":"A114656 \u2192 D+1"},"headline":"A114656 \u2192 D+1","chain":null}},{"id":"l-72f5dea34f7c43a0","kind":"fingerprint_related","node_a_id":"n-7f2a74b39253adec","node_b_id":"n-e09489fb7491a8fa","fingerprint":"f6-48b123f09be537c0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-48b123f09be537c0"},"other_id":"n-7f2a74b39253adec","other":{"id":"n-7f2a74b39253adec","name":"one_plus_derivative 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":"f6-48b123f09be537c0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["6","2"],["20","16","2"]],"card_rows":[["2"],["6","2"],["20","16","2"],["64","96","30","2"]],"row_sums_preview":["2","8","38","192","1002","5336","28814","157184"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a30765140629bd5","steps":["D+1"],"edges":["e-13c630f58cbddced"],"text":"A114655 \u2192 D+1"},"headline":"A114655 \u2192 D+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,      1;\n  2:    10,      8,      1;\n  3:    32,     48,     15,      1;\n  4:    96,    240,    140,     24,      1;\n  5:   272,   1040,   1000,    320,     35,      1;\n  6:   736,   4032,   5880,   3080,    630,     48,     1;\n  7:  1920,  14336,  29792,  23520,   7840,   1120,    63,    1;\n  8:  4864,  47616, 134400, 150528,  75264,  17472,  1848,   80,  1;\n  9: 12032, 149760, 552960, 838656, 592704, 205632, 35280, 2880, 99, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^3+n^2+2*n+8+(2*n^2-8)*x+(n+2)*x^2)*P(n,x) = (2*n^3+6*n^2+12*n+16+(3*n^3+3*n^2+4*n-16)*x+(6*n^2-7*n+4)*x^2+3*n*x^3)*P(n-1,x) + ((4*n^2+2*n+8)*x-(2*n^2-7*n+12)*x^2-(4*n-8)*x^3-2*x^4)*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^3 + 6*n^2 + 12*n + 16 + (3*n^3 + 3*n^2 + 4*n - 16)*x + (6*n^2 - 7*n + 4)*x^2 + 3*n*x^3)*P[n - 1, x] + ((4*n^2 + 2*n + 8)*x - (2*n^2 - 7*n + 12)*x^2 - (4*n - 8)*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 + n^2 + 2*n + 8 + (2*n^2 - 8)*x + (n + 2)*x^2)]];\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^3 + 6*n^2 + 12*n + 16 + (3*n^3 + 3*n^2 + 4*n - 16)*x + (6*n^2 - 7*n + 4)*x^2 + 3*n*x^3)*v[i - 1] + ((4*n^2 + 2*n + 8)*x - (2*n^2 - 7*n + 12)*x^2 - (4*n - 8)*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 + n^2 + 2*n + 8 + (2*n^2 - 8)*x + (n + 2)*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-8ce61d302cec295b","steps":["reverse","D+1"],"edges":["e-0ad5e2ffb4ffd4ba","e-86acb052ed946eeb"],"text":"A114687 \u2192 reverse \u2192 D+1"},"headline":"A114687 \u2192 reverse \u2192 D+1","chain":null}