{"id":"n-f0ea69ccf00996a4","revision":0,"first_index":0,"fingerprint":"f6-12cf750f588dcd1a","name":"theta_plus_one_squared of basis_map_A007318 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k=0 and 0<=k<=2n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["5","20","9"],["25","200","315","160","25"],["125","1500","4050","4400","2250","540","49"],["625","10000","38250","64000","56875","28800","8330","1280","81"],["3125","62500","309375","700000","890625","697500","349125","112000","22275","2500","121"],["15625","375000","2278125","6500000","10781250","11475000","8238125","4080000","1397250","325000","49005","4320","169"],["78125","2187500","15750000","54250000","111015625","149625000","140415625","94600000","46423125","16625000","4298525","781200","94640","6860","225"],["390625","12500000","104062500","420000000","1022656250","1669500000","1937950000","1656000000","1062871875","517500000","191422000","53424000","11061050","1646400","166500","10240","289"],["1953125","70312500","664453125","3075000000","8683593750","16655625000","23056031250","23904000000","18997284375","11735937500","5675731875","2151360000","636158250","145089000","25008750","3148800","273105","14580","361"],["9765625","390625000","4130859375","21562500000","69287109375","152718750000","245650781250","299700000000","284366953125","213296875000","127759740625","61429500000","23732353125","7342650000","1804781250","347520000","51261375","5589000","424175","20000","441"],["48828125","2148437500","25136718750","146093750000","526367187500","1310976562500","2403392578125","3374250000000","3724449609375","3290117187500","2354039875000","1374518250000","657574775000","257945187500","82765546875","21595200000","4536035625","755122500","97289500","9350000","630630","26620","529"],["244140625","11718750000","150292968750","962500000000","3843017578125","10673437500000","21970566406250","34897500000000","43947878906250","44713281250000","37239092343750","25617870000000","14641497203125","6973753500000","2769849843750","915728000000","250882706250","56533950000","10359346250","1518000000","173544525","14907200","904590","34560","625"],["1220703125","63476562500","885498046875","6195312500000","27152099609375","83202539062500","190023818359375","336943750000000","476893828125000","549376953125000","522361348437500","414064462500000","275546195234375","154640891562500","73370288671875","29444584000000","9980987087500","2847970125000","680133025000","134777500000","21890743875","2863646500","294163675","22838400","1259375","43940","729"],["6103515625","341796875000","5152587890625","39101562500000","186621093750000","625447265625000","1569576708984375","3073606250000000","4830812314453125","6217148437500000","6647934261718750","5969691000000000","4537063319140625","2934634040625000","1621310044921875","766598280000000","310345869640625","107454438000000","31734271318750","7957950000000","1683269713125","297525085000","43379256375","5123664000","477750000","33833800","1709505","54880","841"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","34","725","12914","208241","3149146","45518869","636151450","8661161729","115480975346","1513588058741","19556818149154","249647711784625","3153835206933994","39484567407526901"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["25","-25","-30","-5","1"],["0","75","40","-5","-4"],["0","0","25","20","4"]]},"terms":[{"coeff":{"coeffs":[["125","375","450","275","90","15","1"],["0","375","825","700","285","55","4"],["0","0","125","225","145","40","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185842,"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-6039a891c6894b23"]},{"id":185843,"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-6039a891c6894b23"]},{"id":185844,"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-6039a891c6894b23"]}],"region_exact":{},"incoming_edges":[{"id":"e-6039a891c6894b23","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-f0ea69ccf00996a4","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"352346a9241659dd5700a62cee3fe39e5683dc4fd9137428a4f90570e39609f0","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-2ce5b747bc4731d0","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k=0 and 0<=k<=2n.","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, 20, 9;\n 2: 25, 200, 315, 160, 25;\n 3: 125, 1500, 4050, 4400, 2250, 540, 49;\n 4: 625, 10000, 38250, 64000, 56875, 28800, 8330, 1280, 81;\n 5: 3125, 62500, 309375, 700000, 890625, 697500, 349125, 112000, 22275, 2500, 121;\n 6: 15625, 375000, 2278125, 6500000, 10781250, 11475000, 8238125, 4080000, 1397250, 325000, 49005, 4320, 169;\n 7: 78125, 2187500, 15750000, 54250000, 111015625, 149625000, 140415625, 94600000, 46423125, 16625000, 4298525, 781200, 94640, 6860, 225;\n 8: 390625, 12500000, 104062500, 420000000, 1022656250, 1669500000, 1937950000, 1656000000, 1062871875, 517500000, 191422000, 53424000, 11061050, 1646400, 166500, 10240, 289;\n 9: 1953125, 70312500, 664453125, 3075000000, 8683593750, 16655625000, 23056031250, 23904000000, 18997284375, 11735937500, 5675731875, 2151360000, 636158250, 145089000, 25008750, 3148800, 273105, 14580, 361;\n","recurrence_sentence":"The row polynomials satisfy (25+(75*n-25)*x+(25*n^2+40*n-30)*x^2+(20*n^2-5*n-5)*x^3+(4*n^2-4*n+1)*x^4)*P(n,x) = (125+(375*n+375)*x+(125*n^2+825*n+450)*x^2+(225*n^2+700*n+275)*x^3+(145*n^2+285*n+90)*x^4+(40*n^2+55*n+15)*x^5+(4*n^2+4*n+1)*x^6)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((125 + (375*n + 375)*x + (125*n^2 + 825*n + 450)*x^2 + (225*n^2 + 700*n + 275)*x^3 + (145*n^2 + 285*n + 90)*x^4 + (40*n^2 + 55*n + 15)*x^5 + (4*n^2 + 4*n + 1)*x^6)*P[n - 1, x])/(25 + (75*n - 25)*x + (25*n^2 + 40*n - 30)*x^2 + (20*n^2 - 5*n - 5)*x^3 + (4*n^2 - 4*n + 1)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((125 + (375*n + 375)*x + (125*n^2 + 825*n + 450)*x^2 + (225*n^2 + 700*n + 275)*x^3 + (145*n^2 + 285*n + 90)*x^4 + (40*n^2 + 55*n + 15)*x^5 + (4*n^2 + 4*n + 1)*x^6)*v[i - 1])/(25 + (75*n - 25)*x + (25*n^2 + 40*n - 30)*x^2 + (20*n^2 - 5*n - 5)*x^3 + (4*n^2 - 4*n + 1)*x^4)); 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-b30e7a5b4198051a","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-65aaf6e4286453d2","e-6039a891c6894b23"],"text":"A272866 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A272866 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}