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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","1"],["7","10","2"],["33","100","52","6"],["191","1122","1126","316","24"],["1304","14319","24700","12576","2228","120"],["10241","207168","578788","474860","146728","17928","720"],["90865","3372056","14737600","18284740","8750728","1815808","162216","5040"],["898409","61216760","409575260","739107100","513661768","160778296","23934272","1630368","40320"],["9791634","1229048825","12413895780","31753962400","30785316268","13624724680","3002570128","336050640","18019872","362880"],["116601199","27075773730","409125577950","1457066271500","1917862408168","1152425314544","352408361744","57580223392","5018294304","217212480","3628800"],["1506023953","649872133884","14604814843500","71493675884050","125315100935808","99447783154160","40513936467632","9058376033824","1140212162144","79528672224","2835679680","39916800"],["20967734143","16887671994690","562372304081658","3748484506053000","8625542506220528","8861764739322992","4677219525704528","1373830508702944","234118831947424","23384826744576","1334321515008","39856008960","479001600"],["313009988192","472477524261295","23262199388435448","209657509604039504","626516331394528568","820973783943552064","550143843565252368","206420829380637104","45718797402719584","6130262416928320","497476531675968","23643389193984","600037390080","6227020800"],["4987192076929","14161820195678848","1029589086126736184","12482057863966334504","48035755452356389424","79366112293929554464","66503340112209261168","31232614528797460064","8741663560045078784","1510537042015151040","163428678456932480","10984007750431872","441413849488896","9633473579520","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","19","191","2779","55247","1436433","47219053","1910842553","93173743107","5378899667811","362307681907659","28125001045032051","2489818073815623279","249090700567542445377"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","0"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165986,"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-b5244cd67a4007b4"]},{"id":165987,"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-b5244cd67a4007b4"]},{"id":165988,"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-b5244cd67a4007b4"]}],"region_exact":{},"incoming_edges":[{"id":"e-b5244cd67a4007b4","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-5a9cc65a6b56090f","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"9b66eb28f69b67c109ca0353eee64d2e476ea72c97c3166d9e2da695fea9252d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f3e0631eb8740f76","role":"sequence","name":"Triangular array read by rows. 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,          1;\n  2:       7,         10,           2;\n  3:      33,        100,          52,           6;\n  4:     191,       1122,        1126,         316,          24;\n  5:    1304,      14319,       24700,       12576,        2228,         120;\n  6:   10241,     207168,      578788,      474860,      146728,       17928,        720;\n  7:   90865,    3372056,    14737600,    18284740,     8750728,     1815808,     162216,      5040;\n  8:  898409,   61216760,   409575260,   739107100,   513661768,   160778296,   23934272,   1630368,    40320;\n  9: 9791634, 1229048825, 12413895780, 31753962400, 30785316268, 13624724680, 3002570128, 336050640, 18019872, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (n+2+n*x)*P(n-1,x) + n*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (n + 2 + n*x)*P[n - 1, x] + n*x*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; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (n + 2 + n*x)*v[i - 1] + n*x*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-f3e0631eb8740f76","steps":["basis A008277"],"edges":["e-b5244cd67a4007b4"],"text":"A358735 \u2192 basis A008277"},"headline":"A358735 \u2192 basis A008277","chain":null}