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T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","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"],["2","4"],["7","32","32"],["33","240","544","384"],["191","1936","7072","11008","6144"],["1304","17132","87680","218112","263168","122880"],["10241","166416","1105536","3838208","7333888","7299072","2949120"],["90865","1767360","14521792","65280768","173211648","270917632","230916096","82575360"],["898409","20414864","200667904","1113666816","3813767168","8244649984","10975576064","8215068672","2642411520"],["9791634","255138868","2927760256","19409098752","81870232576","227715850240","417307820032","485403918336","324808998912","95126814720"],["116601199","3432916912","45138009888","348905928448","1754992973824","5999154855936","14105352273920","22508944752640","23311222308864","14133881733120","3805072588800"],["1506023953","49503233744","734914712416","6502440406144","38084899278848","154980988125184","446915085402112","912766377918464","1293114650132480","1209339117305856","671356739911680","167423193907200"],["20967734143","761901505248","12619823183648","125961165953536","843550736424960","3991809844281344","13681793273364480","34208144445407232","61893080019632128","78986116966907904","67443466916855808","34569048230461440","8036313307545600"],["313009988192","12469341520412","228181311315200","2539145854446592","19165156776679424","103586804970192896","412354095917694976","1223662650518142976","2705926596805525504","4402022576407183360","5118258729292136448","4025428579939516416","1918135207597178880","417888291992371200"],["4987192076929","216276122160096","4336564283002624","53278525938843136","447991125539454976","2726645849172852736","12384844819740557312","42636223635235799040","111758040897884782592","221885264526657454080","328337960480559595520","350994777071161442304","256106127484860235776","114101784297319956480","23401744351572787200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","71","1201","26351","710276","22702481","839281521","35227121401","1654835424326","86016214943551","4901269416358081","303792687599257471","20349208799733892476","1464839240893241000801"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","0"],["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168068,"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-2162343c24b290d1"]},{"id":168069,"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-2162343c24b290d1"]},{"id":168070,"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-2162343c24b290d1"]},{"id":168071,"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-2162343c24b290d1"]}],"region_exact":{},"incoming_edges":[{"id":"e-2162343c24b290d1","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-8fd05348dab46fd0","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - 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T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","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":[],"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:       2,         4;\n  2:       7,        32,         32;\n  3:      33,       240,        544,         384;\n  4:     191,      1936,       7072,       11008,        6144;\n  5:    1304,     17132,      87680,      218112,      263168,       122880;\n  6:   10241,    166416,    1105536,     3838208,     7333888,      7299072,      2949120;\n  7:   90865,   1767360,   14521792,    65280768,   173211648,    270917632,    230916096,     82575360;\n  8:  898409,  20414864,  200667904,  1113666816,  3813767168,   8244649984,  10975576064,   8215068672,   2642411520;\n  9: 9791634, 255138868, 2927760256, 19409098752, 81870232576, 227715850240, 417307820032, 485403918336, 324808998912, 95126814720;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (n+2+4*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (n + 2 + 4*n*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (n + 2 + 4*n*x)*v[i - 1]); 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-f3e0631eb8740f76","steps":["basis A013611"],"edges":["e-2162343c24b290d1"],"text":"A358735 \u2192 basis A013611"},"headline":"A358735 \u2192 basis A013611","chain":null}