{"id":"n-7f2a74b39253adec","revision":1,"first_index":1,"fingerprint":"f6-48b123f09be537c0","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.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2"],["6","2"],["20","16","2"],["64","96","30","2"],["192","480","280","48","2"],["544","2080","2000","640","70","2"],["1472","8064","11760","6160","1260","96","2"],["3840","28672","59584","47040","15680","2240","126","2"],["9728","95232","268800","301056","150528","34944","3696","160","2"],["24064","299520","1105920","1677312","1185408","411264","70560","5760","198","2"],["58368","901120","4224000","8363520","7983360","3902976","997920","132000","8580","240","2"],["139264","2613248","15178240","38100480","47393280","31223808","11221056","2207040","232320","12320","286","2"],["327680","7348224","51836928","161075200","253693440","217451520","105101568","28993536","4530240","388960","17160","336","2"],["761856","20127744","169594880","639614976","1245916672","1350615552","848060928","314096640","68708640","8744736","624624","23296","390","2"],["1753088","53903360","534876160","2408247296","5689908224","7622078464","6046360320","2905393920","851987136","151575424","16032016","968240","30940","448","2"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","8","38","192","1002","5336","28814","157184","864146","4780008","26572086","148321344","830764794","4666890936","26283115038"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["6","-6","1"],["3","-4","1"],["-2","2","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["8","-20","17","-3"],["6","7","-19","3"],["0","-6","6","0"],["2","3","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","10","-21","12","-2"],["0","-6","11","-4","0"],["0","4","-2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165455,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-c818d14e04074107"]},{"id":165456,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-c818d14e04074107"]},{"id":165457,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-c818d14e04074107"]},{"id":165458,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-c818d14e04074107"]}],"region_exact":{},"incoming_edges":[{"id":"e-c818d14e04074107","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-7f2a74b39253adec","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":"e1fa8f7c98e13b3324738f1fca260657738d62fbd18a3976fe3db404101faafa","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6a30765140629bd5","role":"sequence","name":"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.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-a553f4c52a7a053a","kind":"fingerprint_related","node_a_id":"n-fb038b43f3455c33","node_b_id":"n-7f2a74b39253adec","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-54d5821bb68ac50c"],"text":"A114656 \u2192 D+1"},"headline":"A114656 \u2192 D+1","chain":null}},{"id":"l-08fff8af8583cfbe","kind":"oeis_match","node_a_id":"n-7f2a74b39253adec","node_b_id":null,"fingerprint":null,"oeis_id":"A002003","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A002003","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-67dddcef597b6e34","kind":"oeis_match","node_a_id":"n-7f2a74b39253adec","node_b_id":null,"fingerprint":null,"oeis_id":"A123164","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A123164","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A002003","relation":"match_row_sums"},{"oeis_id":"A123164","relation":"match_row_sums"}],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     2;\n   2:     6,      2;\n   3:    20,     16,       2;\n   4:    64,     96,      30,       2;\n   5:   192,    480,     280,      48,       2;\n   6:   544,   2080,    2000,     640,      70,      2;\n   7:  1472,   8064,   11760,    6160,    1260,     96,     2;\n   8:  3840,  28672,   59584,   47040,   15680,   2240,   126,    2;\n   9:  9728,  95232,  268800,  301056,  150528,  34944,  3696,  160,   2;\n  10: 24064, 299520, 1105920, 1677312, 1185408, 411264, 70560, 5760, 198, 2;\n","recurrence_sentence":"The row polynomials satisfy (n^3-2*n^2+3*n+6+(2*n^2-4*n-6)*x+(n+1)*x^2)*P(n,x) = (2*n^3+6*n+8+(3*n^3-6*n^2+7*n-20)*x+(6*n^2-19*n+17)*x^2+(3*n-3)*x^3)*P(n-1,x) + ((4*n^2-6*n+10)*x-(2*n^2-11*n+21)*x^2-(4*n-12)*x^3-2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.","mathematica":"P[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 6*n + 8 + (3*n^3 - 6*n^2 + 7*n - 20)*x + (6*n^2 - 19*n + 17)*x^2 + (3*n - 3)*x^3)*P[n - 1, x] + ((4*n^2 - 6*n + 10)*x - (2*n^2 - 11*n + 21)*x^2 - (4*n - 12)*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 - 2*n^2 + 3*n + 6 + (2*n^2 - 4*n - 6)*x + (n + 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((2*n^3 + 6*n + 8 + (3*n^3 - 6*n^2 + 7*n - 20)*x + (6*n^2 - 19*n + 17)*x^2 + (3*n - 3)*x^3)*v[i - 1] + ((4*n^2 - 6*n + 10)*x - (2*n^2 - 11*n + 21)*x^2 - (4*n - 12)*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 - 2*n^2 + 3*n + 6 + (2*n^2 - 4*n - 6)*x + (n + 1)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6a30765140629bd5","steps":["D+1"],"edges":["e-c818d14e04074107"],"text":"A114655 \u2192 D+1"},"headline":"A114655 \u2192 D+1","chain":null}
