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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","3"],["7","24","18"],["33","180","306","162"],["191","1452","3978","4644","1944"],["1304","12849","49320","92016","83268","29160"],["10241","124812","621864","1619244","2320488","1732104","524880"],["90865","1325520","8168508","27540324","54805248","64290024","41098104","11022480"],["898409","15311148","112875696","469828188","1206699768","1956494088","1953416736","1096579296","264539520"],["9791634","191354151","1646865144","8188213536","25904253276","54038038680","74271826368","64793601648","32517575712","7142567040"],["116601199","2574687684","25390130562","147194688564","555290745624","1423627568352","2510449660080","3004581431520","2333754418464","1061238075840","214277011200"],["1506023953","37127425308","413389525734","2743217046342","12050300162448","36777714955488","79541283510288","121839603790752","129457477104480","90802848228192","37806457648320","7071141369600"],["20967734143","571426128936","7098650540802","53139866886648","266904725196960","947275187656608","2435065257881520","4566236078009376","6196296661511328","5930647812880128","3797978666280192","1460028454513920","254561089305600"],["313009988192","9352006140309","128351987614800","1071202157344656","6063975386371224","24581634382575072","73390169903320224","163339246623729168","270898199487931104","330524484143915520","288227138238879552","170014523661266688","60759526065626880","9927882482918400"],["4987192076929","162207091620072","2439317409188976","22476878130449448","141747192065218176","647045841161135952","2204236297263395088","5691248845841106720","11188423253342011392","16660185476982874560","18489864567200244480","14824312156158694272","8112507849733925376","2710746214483023360","416971064282572800"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","49","681","12209","267917","6953633","208341073","7076642849","268704087189","11278495019089","518542066790905","25915824844526161","1398935999537621789","81112372148337537601"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","0"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167386,"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-6be4ff430edd1ff3"]},{"id":167387,"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-6be4ff430edd1ff3"]},{"id":167388,"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-6be4ff430edd1ff3"]},{"id":167389,"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-6be4ff430edd1ff3"]}],"region_exact":{},"incoming_edges":[{"id":"e-6be4ff430edd1ff3","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-651bc721fc597ed0","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,         3;\n  2:       7,        24,         18;\n  3:      33,       180,        306,        162;\n  4:     191,      1452,       3978,       4644,        1944;\n  5:    1304,     12849,      49320,      92016,       83268,       29160;\n  6:   10241,    124812,     621864,    1619244,     2320488,     1732104,      524880;\n  7:   90865,   1325520,    8168508,   27540324,    54805248,    64290024,    41098104,    11022480;\n  8:  898409,  15311148,  112875696,  469828188,  1206699768,  1956494088,  1953416736,  1096579296,   264539520;\n  9: 9791634, 191354151, 1646865144, 8188213536, 25904253276, 54038038680, 74271826368, 64793601648, 32517575712, 7142567040;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (n+2+3*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (n + 2 + 3*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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (n + 2 + 3*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 A013610"],"edges":["e-6be4ff430edd1ff3"],"text":"A358735 \u2192 basis A013610"},"headline":"A358735 \u2192 basis A013610","chain":null}