{"id":"n-a27329c52a0854a4","revision":0,"first_index":0,"fingerprint":"f6-be2161e86fda1cad","name":"theta_plus_one of basis_map_A013610 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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"],["4","6"],["32","72","27"],["384","1056","648","108"],["6144","19200","15120","4320","405"],["122880","420864","388800","146880","24300","1458"],["2949120","10838016","11225088","5080320","1134000","122472","5103"],["82575360","321159168","363073536","187149312","50803200","7511616","571536","17496"],["2642411520","10772545536","13063569408","7441072128","2327512320","423263232","44579808","2519424","59049"],["95126814720","403666108416","518764953600","320132874240","111694464000","23616520704","3086294400","243544320","10628820","196830"],["3805072588800","16717405224960","22567095631872","14880374784000","5668276838400","1346760898560","206109598464","20323353600","1247114880","43302600","649539"],["167423193907200","758396265431040","1068180531314688","745004873023488","305205586329600","79663276154880","13782485477376","1600889038848","123464373120","6062364000","171478296","2125764"],["8036313307545600","37407559904133120","54685448697544704","40032956030386176","17443636417658880","4922577366220800","940380769456128","124096909787136","11329290415872","702541382400","28236759408","663238368","6908733"],["417888291992371200","1993410994860195840","3011977351840923648","2300455508370259968","1057192678832209920","318771114147053568","66128820793491456","9677201947066368","1007950172156928","74296453271040","3786698051136","126899607744","2514778812","22320522"],["23401744351572787200","114138345464125194240","177641081179962605568","140873417876098252800","67829498170992230400","21657076036030955520","4818912863950209024","768650694613401600","89105766212136960","7520435290368000","457233386658048","19499208019200","553251338640","9374619240","71744535"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","131","2196","45189","1105182","31354119","1012861224","36717532425","1476342400050","65212709985675","3139386801018300","163605030141437325","9176588125543758150","551225830134140520975"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-16","24","-9"],["32","-24","0"],["-16","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","6"],["8","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190609,"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-32f535c12148514d"]},{"id":190610,"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-32f535c12148514d"]},{"id":190611,"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-32f535c12148514d"]},{"id":190612,"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-32f535c12148514d"]}],"region_exact":{},"incoming_edges":[{"id":"e-32f535c12148514d","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-a27329c52a0854a4","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":"840f9d17ff7a6bcf51d58ca5dda8ceb809f2a6eb46b6e6f3a611e88d6c9de69c","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-a06f8ee83f308dcc","role":"sequence","name":"basis_map_A013610 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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:           4,            6;\n  2:          32,           72,           27;\n  3:         384,         1056,          648,          108;\n  4:        6144,        19200,        15120,         4320,          405;\n  5:      122880,       420864,       388800,       146880,        24300,        1458;\n  6:     2949120,     10838016,     11225088,      5080320,      1134000,      122472,       5103;\n  7:    82575360,    321159168,    363073536,    187149312,     50803200,     7511616,     571536,     17496;\n  8:  2642411520,  10772545536,  13063569408,   7441072128,   2327512320,   423263232,   44579808,   2519424,    59049;\n  9: 95126814720, 403666108416, 518764953600, 320132874240, 111694464000, 23616520704, 3086294400, 243544320, 10628820, 196830;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-32*n+16+(24*n-24)*x+9*x^2)*P(n-2,x) + (8*n-4+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 32*n + 16 + (24*n - 24)*x + 9*x^2)*P[n - 2, x] + (8*n - 4 + 6*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] = 4 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 32*n + 16 + (24*n - 24)*x + 9*x^2)*v[i - 2] + (8*n - 4 + 6*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-9c5c208df8605708","steps":["basis A013610","\u03b8+1"],"edges":["e-bf3b7a758db4c427","e-32f535c12148514d"],"text":"A225471 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A225471 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}