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T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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","3"],["21","33","9"],["102","240","162","27"],["582","1740","1773","702","81"],["3852","13572","17775","10665","2835","243"],["29124","115956","180801","140535","56700","10935","729"],["248112","1087272","1932084","1801548","946890","277263","40824","2187"],["2353968","11153160","21969792","23502528","14904729","5708556","1275750","148716","6561"],["24628320","124566768","266900508","318323412","232834581","108086643","31689630","5603094","531441","19683"],["281797920","1507117968","3464956764","4517375940","3701113965","1990164897","709771167","165009150","23718015","1869885","59049"],["3501325440","19656452736","47998690344","67435719912","60645442770","36669999564","15202051326","4311567711","816254010","97430850","6495390","177147"],["46954235520","275104833408","707903526024","1060342055784","1031354299206","686902311612","321262334217","106225175034","24591763053","3871252440","390372939","22320522","531441"],["676040359680","4114512018432","11087530292592","17559214075728","18267564218904","13199992267176","6820281268515","2551208788557","690260981553","133136658369","17723970693","1531435815","75996063","1594323"],["10402140268800","65512387560960","183944965369104","305975507331312","337529339738136","261595468350216","147038710015197","60918451293843","18690559927182","4222028832021","689790931830","78748041645","5900057982","256686003","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","63","531","4878","48942","534780","6336180","81023760","1113184080","16362954720","256341607200","4264925011200","75119073926400","1396604259187200"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-9"],["0","-6","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","6"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183665,"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-087f8b31a9163994"]},{"id":183666,"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-087f8b31a9163994"]},{"id":183667,"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-087f8b31a9163994"]},{"id":183668,"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-087f8b31a9163994"]}],"region_exact":{},"incoming_edges":[{"id":"e-087f8b31a9163994","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-867c747f4f1fd091","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":"509b728c2d6eb6dce696f35f8ed1092e7da9d3c3e45d14cf5062ca60abb67154","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-c89ee0d57229b178","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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,         3;\n  2:       21,        33,         9;\n  3:      102,       240,       162,        27;\n  4:      582,      1740,      1773,       702,        81;\n  5:     3852,     13572,     17775,     10665,      2835,       243;\n  6:    29124,    115956,    180801,    140535,     56700,     10935,      729;\n  7:   248112,   1087272,   1932084,   1801548,    946890,    277263,    40824,    2187;\n  8:  2353968,  11153160,  21969792,  23502528,  14904729,   5708556,  1275750,  148716,   6561;\n  9: 24628320, 124566768, 266900508, 318323412, 232834581, 108086643, 31689630, 5603094, 531441, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+6*n*x+9*x^2)*P(n-2,x) + (2*n+1+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n*x + 9*x^2)*P[n - 2, x] + (2*n + 1 + 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] = 5 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n*x + 9*x^2)*v[i - 2] + (2*n + 1 + 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-efe23a76953cf69d","steps":["basis A013610","D+1"],"edges":["e-4deb1a58186d6d0a","e-087f8b31a9163994"],"text":"A130534 \u2192 basis A013610 \u2192 D+1"},"headline":"A130534 \u2192 basis A013610 \u2192 D+1","chain":null}