{"id":"n-481b4b083ef06726","revision":1,"first_index":0,"fingerprint":"f6-735f72d4be30f803","name":"T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) = 1.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1","8","4"],["1","22","44","8"],["1","52","264","208","16"],["1","114","1208","2416","912","32"],["1","240","4764","19328","19056","3840","64"],["1","494","17172","124952","249904","137376","15808","128"],["1","1004","58432","705872","2499040","2823488","934912","64256","256"],["1","2026","191360","3641536","20965664","41931328","29132288","6123520","259328","512"],["1","4072","610548","17627904","155809824","503175936","623239296","282046464","39075072","1042432","1024"],["1","8166","1913084","81501480","1061095584","5200393152","10400786304","8488764672","2608047360","244874752","4180992","2048"],["1","16356","5918904","364267600","6772504560","48179888256","145611008256","192719553024","108360072960","23313126400","1515239424","16748544","4096"],["1","32738","18149256","1587286288","41147874800","410984411616","1790155269888","3580310539776","3287875292928","1316731993600","203172644864","9292419072","67047424","8192"],["1","65504","55298956","6784727296","240659672336","3288863965696","19928806298304","57284968636416","79715225193216","52621823451136","15402219029504","1736890187776","56626130944","268304384","16384"],["1","131038","167730980","28576682040","1366131816784","25027696585568","205278923402048","816998262460800","1633996524921600","1642231387216384","800886290738176","174864872548352","14631261204480","343513047040","1073463296","32768"],["1","262108","507124080","119003195600","7573652816960","182953343116224","1987561788696832","10641564567727360","29411937448588800","42566258270909440","31800988619149312","11709013959438336","1938855121141760","121859272294400","2077180231680","4294377472","65536"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["13","75","541","4683","47293","545835","7087261","102247563","1622632573","28091567595","526858348381","10641342970443","230283190977853","5315654681981355","130370767029135901"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A156365","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","8","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","4"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159446,"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":false,"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-4e3e568f497fe445"]},{"id":159447,"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-4e3e568f497fe445"]},{"id":159448,"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-4e3e568f497fe445"]},{"id":159449,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-4e3e568f497fe445"]}],"region_exact":{},"incoming_edges":[{"id":"e-4e3e568f497fe445","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-481b4b083ef06726","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"770a20c54e378a7a73736f2a88fb61bf8d49315c1db154b3d8c60d254a0925f3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-00ad498f5c132fdf","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-c0390a92c6c5f4ab","conclusion_name":"obreschkoff_combination of T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) = 1."},{"id":"e-00ad498f5c132fdf","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-c0390a92c6c5f4ab","conclusion_name":"obreschkoff_combination of T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) = 1."}],"used_by_total":2,"links":[{"id":"l-70775ff3f27eebaf","kind":"fingerprint_related","node_a_id":"n-423e26f84d00b286","node_b_id":"n-481b4b083ef06726","fingerprint":"f6-735f72d4be30f803","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-735f72d4be30f803"},"other_id":"n-423e26f84d00b286","other":{"id":"n-423e26f84d00b286","name":"Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.","first_index":1,"fingerprint":"f6-735f72d4be30f803","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","8","4"]],"card_rows":[["1"],["1","2"],["1","8","4"],["1","22","44","8"]],"row_sums_preview":["1","3","13","75","541","4683","47293","545835"],"regions":["(-inf, 0)"],"region_exact":{},"oeis_ids":["A142075"],"derivation":{"root":"n-423e26f84d00b286","steps":[],"edges":[],"text":"A142075"},"headline":"A142075: Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.","chain":null}},{"id":"l-e7fca3b364668e45","kind":"oeis_match","node_a_id":"n-481b4b083ef06726","node_b_id":null,"fingerprint":null,"oeis_id":"A000670","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000670","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A156365","relation":"identification"},{"oeis_id":"A000670","relation":"match_row_sums"}],"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,    8,       4;\n  1: 1,   22,      44,        8;\n  2: 1,   52,     264,      208,         16;\n  3: 1,  114,    1208,     2416,        912,         32;\n  4: 1,  240,    4764,    19328,      19056,       3840,          64;\n  5: 1,  494,   17172,   124952,     249904,     137376,       15808,        128;\n  6: 1, 1004,   58432,   705872,    2499040,    2823488,      934912,      64256,        256;\n  7: 1, 2026,  191360,  3641536,   20965664,   41931328,    29132288,    6123520,     259328,       512;\n  8: 1, 4072,  610548, 17627904,  155809824,  503175936,   623239296,  282046464,   39075072,   1042432,    1024;\n  9: 1, 8166, 1913084, 81501480, 1061095584, 5200393152, 10400786304, 8488764672, 2608047360, 244874752, 4180992, 2048;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(2*n+4)*x)*P(n-1,x) + (x-2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 8*x + 4*x^2.","mathematica":"P[0, x_] := 1 + 8*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (2*n + 4)*x)*P[n - 1, x] + (x - 2*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 8*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (2*n + 4)*x)*v[i - 1] + (x - 2*x^2)*deriv(v[i - 1], x)); 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-481b4b083ef06726","steps":[],"edges":[],"text":"A156365"},"headline":"A156365: T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) = 1.","chain":null}