{"id":"n-624a28504ef54c1f","revision":0,"first_index":0,"fingerprint":"f6-66ed502267415f23","name":"theta_plus_one of basis_map_A013609 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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"],["10","52","48"],["100","880","1884","1152"],["1700","21360","74268","99968","46080"],["44200","725360","3532968","7550368","7446080","2764800"],["1635400","33203120","209058696","618528544","955038080","745638912","232243200"],["81770000","1980694400","15333793440","58243430144","123521650640","149594423808","96863542272","26011238400"],["5315050000","149678256000","1376989898400","6402790373120","17347856114640","28696563085824","28635564203008","15860482965504","3745618329600"],["435834100000","13995693192000","149284987876800","822458628650240","2719089251957280","5725541401723584","7771766687867392","6602229775613952","3197713723490304","674211299328000"],["44019244100000","1587898652392000","19276491733156800","122877651594154240","480242671610245280","1230861102043829184","2120908095876776192","2443229021706698752","1808470785585659904","778698390896640000","148326485852160000"],["5370347780200000","215028949736224000","2928139202263425600","21210954827052078080","95760095543681581760","289627526284962389760","606263905507341135040","884656522593565634560","885802537001099243520","581278903813534187520","225385342930649088000","39158192264970240000"],["778700428129000000","34272518033147680000","517472690614245480000","4199993903591706752000","21521157591572577464000","75090680331215901171456","185223115130311827540160","327822629817340442030080","415070081185474655907840","367742252893314215116800","216830031453071748366336","76489914711904616448000","12217355986670714880000"],["132379072781930000000","6352729555050309600000","105346524047227605360000","947206656180743444160000","5433094214834834271600000","21494397175448540618545920","61098687849429147710438016","127278890399285472526341120","195216468789574394586270720","218398057925963898455244800","173587674972756422402703360","92954431644083694921056256","30085007621409761918976000","4447117579148140216320000"],["26078677338040210000000","1355272915405944111200000","24488670215673420300720000","241660827836290753567680000","1534450821851026639143600000","6790123762221668543214186240","21866545814576008008504623232","52446153565203496261965431808","94587635017631049117611765760","128052034928651735415481139200","128370014547308648227234119680","92544425131325052534204137472","45401395139605265541180162048","13576583842038089269641216000","1867789383242218890854400000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","110","4016","243376","22063776","2795345952","471630543104","102220858276096","27676727283799552","9154522614795812352","3632292519266635427840","1702759736913543927836672","931052866460790110916616192","587304682803298966607224381440"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","4"]],"recurrence":{"denominator":{"coeffs":[["1"],["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","0"],["-4","-8","-4"],["8","24","16"],["-8","-28","-24"],["4","16","16"],["-1","-4","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","0"],["-2","0"],["4","4"],["-4","-8"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193675,"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-00f005c4d40ed497"]},{"id":193676,"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-00f005c4d40ed497"]},{"id":193677,"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-00f005c4d40ed497"]},{"id":193678,"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-00f005c4d40ed497"]}],"region_exact":{},"incoming_edges":[{"id":"e-00f005c4d40ed497","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-624a28504ef54c1f","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":"1fd4b1c7f24560a71e5ff402fc821dd98f0ca548760fc05a114de564170a1a7e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9116f92e53f934a9","role":"sequence","name":"basis_map_A013609 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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:           10,             52,              48;\n  3:          100,            880,            1884,            1152;\n  4:         1700,          21360,           74268,           99968,            46080;\n  5:        44200,         725360,         3532968,         7550368,          7446080,          2764800;\n  6:      1635400,       33203120,       209058696,       618528544,        955038080,        745638912,        232243200;\n  7:     81770000,     1980694400,     15333793440,     58243430144,     123521650640,     149594423808,      96863542272,      26011238400;\n  8:   5315050000,   149678256000,   1376989898400,   6402790373120,   17347856114640,   28696563085824,   28635564203008,   15860482965504,    3745618329600;\n  9: 435834100000, 13995693192000, 149284987876800, 822458628650240, 2719089251957280, 5725541401723584, 7771766687867392, 6602229775613952, 3197713723490304, 674211299328000;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(4*n^6-16*n^5+28*n^4-24*n^3+8*n^2)*x+(4*n^6-16*n^5+24*n^4-16*n^3+4*n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(4*n^4-8*n^3+4*n^2)*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[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (4*n^6 - 16*n^5 + 28*n^4 - 24*n^3 + 8*n^2)*x + (4*n^6 - 16*n^5 + 24*n^4 - 16*n^3 + 4*n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (4*n^4 - 8*n^3 + 4*n^2)*x)*P[n - 1, x])/(n^2 - 2*n + 1)]];\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] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (4*n^6 - 16*n^5 + 28*n^4 - 24*n^3 + 8*n^2)*x + (4*n^6 - 16*n^5 + 24*n^4 - 16*n^3 + 4*n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (4*n^4 - 8*n^3 + 4*n^2)*x)*v[i - 1])/(n^2 - 2*n + 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-e734a631f8279507","steps":["basis A013609","\u03b8+1"],"edges":["e-031a48225080e193","e-00f005c4d40ed497"],"text":"A008955 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A008955 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}