{"id":"n-1cc5242f219c3841","revision":1,"first_index":1,"fingerprint":"f6-8efdaeff5d010efc","name":"derivative of Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["2"],["3","6"],["4","32","12"],["5","110","165","20"],["6","312","1188","624","30"],["7","798","6342","8456","1995","42"],["8","1920","28584","77312","47640","5760","56"],["9","4446","115911","562284","702855","231822","15561","72"],["10","10040","438240","3529360","7809500","5294040","1022560","40160","90"],["11","22286","1578720","20028448","72069470","86483364","35049784","4209920","100287","110"],["12","48864","5494932","105767424","584286840","1132145856","818001576","211534848","16484796","244320","132"],["13","106158","18652569","529759620","4310700810","12675958308","14788618026","6897121296","1191959145","62175230","583869","156"],["14","228984","62148492","2549873200","29629707450","126472206672","222966856392","168629608896","53333473410","6374683000","227877804","1373904","182"]],"degrees":[-1,-1,0,1,2,3,4,5,6,7,8,9,10,11,12],"row_sums":["0","0","2","9","48","300","2160","17640","161280","1632960","18144000","219542400","2874009600","40475635200","610248038400"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["6"],["-5"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","-16"],["0","0","36"],["0","0","-28"],["0","0","9"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","4","-4"],["0","0","-8","8"],["0","0","5","-5"],["0","0","-1","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","-18"],["-4","30"],["1","-14"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"],["0","-4","4"],["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164707,"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-5bbae1ad83a2b739"]},{"id":164708,"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-5bbae1ad83a2b739"]},{"id":164709,"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-5bbae1ad83a2b739"]},{"id":164710,"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-5bbae1ad83a2b739"]}],"region_exact":{},"incoming_edges":[{"id":"e-5bbae1ad83a2b739","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-1cc5242f219c3841","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":"f94c748c1336ae12c99644628e53ef74fbcc11fc552263e8bd838d9492022380","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-3ffff24fcb4f40a0","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","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-444c5e8b74d1f5d4","kind":"fingerprint_related","node_a_id":"n-281d59af1fd93227","node_b_id":"n-1cc5242f219c3841","fingerprint":"f6-8efdaeff5d010efc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8efdaeff5d010efc"},"other_id":"n-281d59af1fd93227","other":{"id":"n-281d59af1fd93227","name":"theta_plus_one of Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.","first_index":1,"fingerprint":"f6-8efdaeff5d010efc","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","2"]],"card_rows":[["1"],["1"],["1","2"],["1","8","3"]],"row_sums_preview":["1","1","3","12","60","360","2520","20160"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efd1fdf07e5468d8","steps":["\u03b8+1"],"edges":["e-6ce185e3aebdd3d4"],"text":"A173018 \u2192 \u03b8+1"},"headline":"A173018 \u2192 \u03b8+1","chain":null}},{"id":"l-7546a7f2a99e29ba","kind":"fingerprint_related","node_a_id":"n-4f741935109d06a2","node_b_id":"n-1cc5242f219c3841","fingerprint":"f6-8efdaeff5d010efc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8efdaeff5d010efc"},"other_id":"n-4f741935109d06a2","other":{"id":"n-4f741935109d06a2","name":"derivative of Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-8efdaeff5d010efc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["1","2"]],"card_rows":[["1"],["1","2"],["1","8","3"],["1","22","33","4"]],"row_sums_preview":["0","1","3","12","60","360","2520","20160"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff270bcb802a85af","steps":["\u03b8+1"],"edges":["e-6afbce5b42114ab6"],"text":"A008292 \u2192 \u03b8+1"},"headline":"A008292 \u2192 \u03b8+1","chain":null}},{"id":"l-a377122684e08162","kind":"fingerprint_related","node_a_id":"n-8293244ef059e34c","node_b_id":"n-1cc5242f219c3841","fingerprint":"f6-8efdaeff5d010efc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8efdaeff5d010efc"},"other_id":"n-8293244ef059e34c","other":{"id":"n-8293244ef059e34c","name":"Triangle with T(n,k) = k*E(n,k) where E(n,k) are Eulerian numbers A008292.","first_index":1,"fingerprint":"f6-8efdaeff5d010efc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1","8","3"],["1","22","33","4"],["1","52","198","104","5"]],"card_rows":[["1","8","3"],["1","22","33","4"],["1","52","198","104","5"],["1","114","906","1208","285","6"]],"row_sums_preview":["12","60","360","2520","20160","181440","1814400","19958400"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff270bcb802a85af","steps":["\u03b8+1 (shift +2)"],"edges":["e-fb10cc40ca2d4795"],"text":"A008292 \u2192 \u03b8+1 (shift +2)"},"headline":"A008292 \u2192 \u03b8+1 (shift +2)","chain":null}},{"id":"l-10b1a024a2559142","kind":"oeis_match","node_a_id":"n-1cc5242f219c3841","node_b_id":null,"fingerprint":null,"oeis_id":"A074143","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":3,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A074143","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A074143","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: 0;\n   2: 0;\n   3: 2;\n   4: 3,    6;\n   5: 4,   32,     12;\n   6: 5,  110,    165,     20;\n   7: 6,  312,   1188,    624,     30;\n   8: 7,  798,   6342,   8456,   1995,     42;\n   9: 8, 1920,  28584,  77312,  47640,   5760,    56;\n  10: 9, 4446, 115911, 562284, 702855, 231822, 15561, 72;\n","recurrence_sentence":"The row polynomials satisfy (n^2-5*n+6)*P(n,x) = -(n^4-9*n^3+28*n^2-36*n+16)*x^2*P(n-2,x) - ((n^3-5*n^2+8*n-4)*x^2-(n^3-5*n^2+8*n-4)*x^3)*d/dx P(n-2,x) + (n^2-4*n+3+(2*n^3-14*n^2+30*n-18)*x)*P(n-1,x) + ((n^2-4*n+3)*x-(n^2-4*n+3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - 9*n^3 + 28*n^2 - 36*n + 16)*x^2*P[n - 2, x] - ((n^3 - 5*n^2 + 8*n - 4)*x^2 - (n^3 - 5*n^2 + 8*n - 4)*x^3)*D[P[n - 2, x], x] + (n^2 - 4*n + 3 + (2*n^3 - 14*n^2 + 30*n - 18)*x)*P[n - 1, x] + ((n^2 - 4*n + 3)*x - (n^2 - 4*n + 3)*x^2)*D[P[n - 1, x], x])/(n^2 - 5*n + 6)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 4, N, my(i = n); v[i] = (-(n^4 - 9*n^3 + 28*n^2 - 36*n + 16)*x^2*v[i - 2] - ((n^3 - 5*n^2 + 8*n - 4)*x^2 - (n^3 - 5*n^2 + 8*n - 4)*x^3)*deriv(v[i - 2], x) + (n^2 - 4*n + 3 + (2*n^3 - 14*n^2 + 30*n - 18)*x)*v[i - 1] + ((n^2 - 4*n + 3)*x - (n^2 - 4*n + 3)*x^2)*deriv(v[i - 1], x))/(n^2 - 5*n + 6)); 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-3ffff24fcb4f40a0","steps":["D"],"edges":["e-5bbae1ad83a2b739"],"text":"A334218 \u2192 D"},"headline":"A334218 \u2192 D","chain":null}
