{"id":"n-3404697af6c67a78","revision":0,"first_index":0,"fingerprint":"f6-858fd07a4a9f264c","name":"basis_map_A008277 of reversal of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","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"],["5","2"],["18","18","3"],["56","106","42","4"],["160","515","365","80","5"],["432","2241","2535","960","135","6"],["1120","9086","15442","8995","2135","210","7"],["2816","35092","86380","72688","25900","4228","308","8"],["6912","130869","455499","532602","267309","64386","7686","432","9"],["16640","475435","2301525","3642880","2474745","824880","143430","13080","585","10"],["39424","1692680","11263736","23690645","21202115","9393384","2230767","293370","21120","770","11"],["92160","5930838","53791914","148305432","171486810","98241330","30598722","5440644","560340","32670","990","12"],["212992","20513207","252010057","901434820","1327313273","963846312","382930834","88386870","12208911","1011725","48763","1248","13"],["487424","70194229","1162687435","5353240984","9926067151","9000629638","4470173708","1302489188","231660429","25578553","1742741","70616","1547","14"],["1105920","238042530","5297885790","31204002375","72225473775","80835384630","49427172795","17814139200","3966924390","560269710","50585535","2884245","99645","1890","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","39","208","1125","6309","36995","227420","1465704","9893210","69828022","514481862","3949919025","31545023657","261623972445"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","2"],["18","18","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["8","19","9","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","19","18","3"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","9","3"]]},"deriv_order":2,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","1"]]},"deriv_order":3,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-12","-15","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-15","-6"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-3"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":196306,"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-ba6433a727aa9661"]},{"id":196307,"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-ba6433a727aa9661"]},{"id":196308,"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-ba6433a727aa9661"]}],"region_exact":{},"incoming_edges":[{"id":"e-ba6433a727aa9661","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-3404697af6c67a78","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":"3e4214817e4e442970a551a977d18adcdd80acf3492bcb5d8cd90c2564a6fca5","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-2c11b65ec5b65118","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","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:     5,      2;\n  2:    18,     18,       3;\n  3:    56,    106,      42,       4;\n  4:   160,    515,     365,      80,       5;\n  5:   432,   2241,    2535,     960,     135,      6;\n  6:  1120,   9086,   15442,    8995,    2135,    210,      7;\n  7:  2816,  35092,   86380,   72688,   25900,   4228,    308,     8;\n  8:  6912, 130869,  455499,  532602,  267309,  64386,   7686,   432,   9;\n  9: 16640, 475435, 2301525, 3642880, 2474745, 824880, 143430, 13080, 585, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+19*x+9*x^2+x^3)*P(n-3,x) + (19*x+18*x^2+3*x^3)*d/dx P(n-3,x) + (9*x^2+3*x^3)*d^2/dx^2 P(n-3,x) + x^3*d^3/dx^3 P(n-3,x) - (12+15*x+3*x^2)*P(n-2,x) - (15*x+6*x^2)*d/dx P(n-2,x) - 3*x^2*d^2/dx^2 P(n-2,x) + (6+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 2*x, P(2,x) = 18 + 18*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 2*x;\nP[2, x_] := 18 + 18*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[(8 + 19*x + 9*x^2 + x^3)*P[n - 3, x] + (19*x + 18*x^2 + 3*x^3)*D[P[n - 3, x], x] + (9*x^2 + 3*x^3)*D[P[n - 3, x], {x, 2}] + x^3*D[P[n - 3, x], {x, 3}] - (12 + 15*x + 3*x^2)*P[n - 2, x] - (15*x + 6*x^2)*D[P[n - 2, x], x] - 3*x^2*D[P[n - 2, x], {x, 2}] + (6 + 3*x)*P[n - 1, x] + 3*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] = 5 + 2*x; v[3] = 18 + 18*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 19*x + 9*x^2 + x^3)*v[i - 3] + (19*x + 18*x^2 + 3*x^3)*deriv(v[i - 3], x) + (9*x^2 + 3*x^3)*deriv(deriv(v[i - 3], x), x) + x^3*deriv(deriv(deriv(v[i - 3], x), x), x) - (12 + 15*x + 3*x^2)*v[i - 2] - (15*x + 6*x^2)*deriv(v[i - 2], x) - 3*x^2*deriv(deriv(v[i - 2], x), x) + (6 + 3*x)*v[i - 1] + 3*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-787407680a56b31c","steps":["reverse","basis A008277"],"edges":["e-03e6ee68811b0a8b","e-ba6433a727aa9661"],"text":"A124932 \u2192 reverse \u2192 basis A008277"},"headline":"A124932 \u2192 reverse \u2192 basis A008277","chain":null}
