{"id":"n-6d28c1bea5c74132","revision":0,"first_index":0,"fingerprint":"f6-8b81ab23f79e321a","name":"derivative of basis_map_A013610 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["3"],["36","54"],["1035","2430","1215"],["55440","162540","136080","34020"],["4788315","16159500","17605350","7654500","1148175"],["608855940","2271328290","2930142600","1700829900","454677300","45467730"],["107110838835","431345244330","628066804365","439807668300","159755921325","28962944010","2068781715"],["24921704808000","106692655469400","170430834974400","136640733629400","60684263640000","15143482153800","1986030446400","106394488200"],["7411923037393875","33376191321072600","57416499451010700","51078468797155800","26248598960834250","8090343277216200","1474999987160700","146505210251400","6104383760475"],["2743388978865034500","12893764293170110350","23581020659766570000","22802981194050057000","13115085126658515000","4716018091996110900","1070482823299890000","148974094350081000","11598329144902500","386610971496750"],["1236840759863452618875","6031817064625013011350","11617690836661632798075","12036920441996467017000","7579487309887847103750","3071264414145668640900","818710254836965641150","142772338885977801000","15673402089963993375","982378478573241750","26792140324724775"],["667332766316625106698000","3361515252047788344408900","6770051518782692596856400","7436358470789001269415900","5047733175052866398100000","2252280424341772451226600","680334510147481149760800","140087718794968690145400","19373932511614979298000","1721305708729151044500","88735568755488454800","2016717471715646700"],["424583281448484427691296875","2201046947477361481260262500","4609452665094559061781251250","5324729196121395053070577500","3851991841044510367820240625","1861971250968365757845625000","622283605972080808082827500","145869210830925244177515000","23928609093084845292268125","2690310443923634344312500","197467451442335245421250","8520631317998607307500","163858294576896294375"],["314583916363845919590362062500","1673345716305650997124721456250","3627718196972432976011654055000","4379548471802709372360826132500","3347230883228739307264351387500","1731790648905465045221588013750","629601031970157473208792810000","163944337077159983525075895000","30736287865826915093739517500","4114811131997965513307943750","383660432655058213990335000","23660482303725517322572500","867138094900935189832500","14293485080784645986250"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","90","4680","388080","47355840","8011301760","1797118202880","516606099609600","185249634425856000","81083700202272768000","42552185993611886592000","26376879826779767341056000","19068753070435745592115200000","15903022503970524720365568000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-27","54","-27"],["108","-180","72"],["-144","168","-36"],["80","-48","0"],["-16","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","-6"],["-12","12"],["8","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194163,"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-d0df31198dc15c6e"]},{"id":194164,"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-d0df31198dc15c6e"]},{"id":194165,"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-d0df31198dc15c6e"]},{"id":194166,"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-d0df31198dc15c6e"]}],"region_exact":{},"incoming_edges":[{"id":"e-d0df31198dc15c6e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-6d28c1bea5c74132","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"a548664803ae94e257965300def421cdfe4b338ca5542f4014b3702da5487773","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-77311ba2b23da2eb","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..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:                0;\n  1:                3;\n  2:               36,                54;\n  3:             1035,              2430,              1215;\n  4:            55440,            162540,            136080,             34020;\n  5:          4788315,          16159500,          17605350,           7654500,           1148175;\n  6:        608855940,        2271328290,        2930142600,        1700829900,         454677300,         45467730;\n  7:     107110838835,      431345244330,      628066804365,      439807668300,      159755921325,      28962944010,       2068781715;\n  8:   24921704808000,   106692655469400,   170430834974400,   136640733629400,    60684263640000,   15143482153800,    1986030446400,    106394488200;\n  9: 7411923037393875, 33376191321072600, 57416499451010700, 51078468797155800, 26248598960834250, 8090343277216200, 1474999987160700, 146505210251400, 6104383760475;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(48*n^3-168*n^2+180*n-54)*x+(36*n^2-72*n+27)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(12*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (48*n^3 - 168*n^2 + 180*n - 54)*x + (36*n^2 - 72*n + 27)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (12*n - 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] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (48*n^3 - 168*n^2 + 180*n - 54)*x + (36*n^2 - 72*n + 27)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (12*n - 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-1336d058dca9979e","steps":["basis A013610","D"],"edges":["e-05481f5802c9cdfa","e-d0df31198dc15c6e"],"text":"A204420 \u2192 basis A013610 \u2192 D"},"headline":"A204420 \u2192 basis A013610 \u2192 D","chain":null}