{"id":"n-8293244ef059e34c","revision":1,"first_index":1,"fingerprint":"f6-8efdaeff5d010efc","name":"Triangle with T(n,k) = k*E(n,k) where E(n,k) are Eulerian numbers A008292.","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":[["1","8","3"],["1","22","33","4"],["1","52","198","104","5"],["1","114","906","1208","285","6"],["1","240","3573","9664","5955","720","7"],["1","494","12879","62476","78095","25758","1729","8"],["1","1004","43824","352936","780950","529404","102256","4016","9"],["1","2026","143520","1820768","6551770","7862124","3186344","382720","9117","10"],["1","4072","457911","8813952","48690570","94345488","68166798","17627904","1373733","20360","11"],["1","8166","1434813","40750740","331592370","975073716","1137586002","530547792","91689165","4782710","44913","12"],["1","16356","4439178","182133800","2116407675","9033729048","15926204028","12044972064","3809533815","455334500","16276986","98136","13"],["1","32738","13611942","793643144","12858710875","77059577178","195798232644","223769408736","115589365767","25717421750","2182518646","54447768","212797","14"],["1","65504","41474217","3392363648","75206147605","616661993568","2179713188877","3580310539776","2802488385699","1027769989280","165453524731","10177090944","179721607","458528","15"],["1","131038","125798235","14288341020","426916192745","4692693109794","22452382247099","51062391403800","57445190329275","32074831781570","8603270701289","1024598862588","46437108315","587058430","982785","16"],["1","262108","380343060","59501597800","2366766505300","34303751834292","217389570638716","665097785482960","1034013425926950","831372231853700","341612182432268","68607503668584","6153592913780","208255592300","1901715300","2096864","17"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["12","60","360","2520","20160","181440","1814400","19958400","239500800","3113510400","43589145600","653837184000","10461394944000","177843714048000","3201186852864000"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A065826","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1","8","3"],["1","22","33","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","-2"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2","2"],["0","0","-1","1"]]},"deriv_order":1,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160078,"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-fb10cc40ca2d4795"]},{"id":160079,"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-fb10cc40ca2d4795"]},{"id":160080,"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-fb10cc40ca2d4795"]},{"id":160081,"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-fb10cc40ca2d4795"]}],"region_exact":{},"incoming_edges":[{"id":"e-fb10cc40ca2d4795","operation":{"index_shift":2,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 2","conclusion_id":"n-8293244ef059e34c","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"61120baf09fcd1abc4cc82ea0a697ac908f0999421fa467e2ab050b5b759d7ee","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-ff270bcb802a85af","role":"sequence","name":"eulerian","uses":{"claim":"real_rooted","from_index":3,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 3, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-45c6baa8c86f1355","kind":"fingerprint_related","node_a_id":"n-8293244ef059e34c","node_b_id":"n-4f741935109d06a2","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-1cc5242f219c3841","other":{"id":"n-1cc5242f219c3841","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.","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":[[],[],["2"]],"card_rows":[["2"],["3","6"],["4","32","12"],["5","110","165","20"]],"row_sums_preview":["0","0","2","9","48","300","2160","17640"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3ffff24fcb4f40a0","steps":["D"],"edges":["e-5bbae1ad83a2b739"],"text":"A334218 \u2192 D"},"headline":"A334218 \u2192 D","chain":null}},{"id":"l-b4f93a4737581a77","kind":"fingerprint_related","node_a_id":"n-8293244ef059e34c","node_b_id":"n-281d59af1fd93227","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-42f87a0e67e7e9ef","kind":"oeis_match","node_a_id":"n-8293244ef059e34c","node_b_id":null,"fingerprint":null,"oeis_id":"A188588","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":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A188588","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-63b2376bea3a36a5","kind":"oeis_match","node_a_id":"n-8293244ef059e34c","node_b_id":null,"fingerprint":null,"oeis_id":"A357711","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":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A357711","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A188588","relation":"match_row_sums"},{"oeis_id":"A357711","relation":"match_row_sums"},{"oeis_id":"A065826","relation":"tail"}],"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: 1,    8,       3;\n   2: 1,   22,      33,        4;\n   3: 1,   52,     198,      104,         5;\n   4: 1,  114,     906,     1208,       285,         6;\n   5: 1,  240,    3573,     9664,      5955,       720,          7;\n   6: 1,  494,   12879,    62476,     78095,     25758,       1729,         8;\n   7: 1, 1004,   43824,   352936,    780950,    529404,     102256,      4016,        9;\n   8: 1, 2026,  143520,  1820768,   6551770,   7862124,    3186344,    382720,     9117,      10;\n   9: 1, 4072,  457911,  8813952,  48690570,  94345488,   68166798,  17627904,  1373733,   20360,    11;\n  10: 1, 8166, 1434813, 40750740, 331592370, 975073716, 1137586002, 530547792, 91689165, 4782710, 44913, 12;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(2*n+2)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) - (n^2+2*n)*x^2*P(n-2,x) - ((n+2)*x^2-(n+2)*x^3)*d/dx P(n-2,x) for n >= 3, with P(1,x) = 1 + 8*x + 3*x^2, P(2,x) = 1 + 22*x + 33*x^2 + 4*x^3.","mathematica":"P[1, x_] := 1 + 8*x + 3*x^2;\nP[2, x_] := 1 + 22*x + 33*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[(1 + (2*n + 2)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x] - (n^2 + 2*n)*x^2*P[n - 2, x] - ((n + 2)*x^2 - (n + 2)*x^3)*D[P[n - 2, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 8*x + 3*x^2; v[2] = 1 + 22*x + 33*x^2 + 4*x^3; for(n = 3, N, my(i = n); v[i] = (1 + (2*n + 2)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x) - (n^2 + 2*n)*x^2*v[i - 2] - ((n + 2)*x^2 - (n + 2)*x^3)*deriv(v[i - 2], x)); 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-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}