{"id":"n-93c4bc52c48c6619","revision":0,"first_index":0,"fingerprint":"f6-7c9f7ca6f2f9d8f8","name":"one_plus_derivative of basis_map_A013609 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","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"],["3","2"],["13","26","8"],["83","268","244","48"],["701","3084","4652","2672","384"],["7359","40726","85148","81928","34016","3840"],["92289","613262","1631212","2194952","1535104","496512","46080"],["1345287","10410968","33511000","57743808","56774896","31026560","8195328","645120"],["22341465","197051544","742847016","1556522944","1967952144","1516990848","678226688","151123968","10321920"],["416291355","4117682538","17765961000","43759007120","67495234992","67092083616","42406644736","16011062784","3081498624","185794560"],["8598724245","94198433922","457265415840","1292727259920","2349887916688","2856175566624","2332621321344","1247068729856","406956681216","68875960320","3715891200"],["194977248795","2342106939492","12623738105196","40244125047536","84138392538064","120746578585792","120781994371648","83557053098240","38654673781760","11097217216512","1675054448640","81749606400"],["4814460176085","62897384096676","372487090982580","1320930979454512","3119420182592432","5156681381954240","6098858628688832","5174267656584448","3100886984332288","1263087273160704","323478876192768","44043538268160","1961990553600"],["128583592125975","1814564443849470","11706698315907684","45683097288815704","120171971787855984","224674491894294304","306448712203493568","307736557730584704","226393310817144320","119565008361101312","43474791423270912","10044530179964928","1245193513205760","51011754393600"],["3693120887658825","55972557242732790","390603072831550308","1662588892633824984","4818212215133064752","10045101082663084576","15511136199465078464","17980083678247508096","15669991875848327168","10167075021468774400","4796440916666335232","1574177642503766016","331164209666064384","37671064861409280","1428329123020800"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","47","643","11493","253017","6609411","199652967","6843378537","262331261325","11118091901175","516137660988075","26043816427037325","1419138523306008225","83045339879242200075"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","2"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["-9","-12","-4"],["12","20","8"],["-4","-8","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","0"],["-6","-4"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187875,"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-963be6917ea50d69"]},{"id":187876,"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-963be6917ea50d69"]},{"id":187877,"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-963be6917ea50d69"]},{"id":187878,"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-963be6917ea50d69"]}],"region_exact":{},"incoming_edges":[{"id":"e-963be6917ea50d69","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-93c4bc52c48c6619","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":"ce927a1366b5989915d64ea49b9899cfb7fb31be8b7f0d6e7ef9ec39c3d88019","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-1e0cd3ca23b1a82f","role":"sequence","name":"basis_map_A013609 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","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:         3,          2;\n  2:        13,         26,           8;\n  3:        83,        268,         244,          48;\n  4:       701,       3084,        4652,        2672,         384;\n  5:      7359,      40726,       85148,       81928,       34016,        3840;\n  6:     92289,     613262,     1631212,     2194952,     1535104,      496512,       46080;\n  7:   1345287,   10410968,    33511000,    57743808,    56774896,    31026560,     8195328,      645120;\n  8:  22341465,  197051544,   742847016,  1556522944,  1967952144,  1516990848,   678226688,   151123968,   10321920;\n  9: 416291355, 4117682538, 17765961000, 43759007120, 67495234992, 67092083616, 42406644736, 16011062784, 3081498624, 185794560;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(8*n^3-20*n^2+12*n)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (4*n^2-6*n+1+(4*n^2-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (8*n^3 - 20*n^2 + 12*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (4*n^2 - 6*n + 1 + (4*n^2 - 4*n)*x)*P[n - 1, x])/(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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (8*n^3 - 20*n^2 + 12*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (4*n^2 - 6*n + 1 + (4*n^2 - 4*n)*x)*v[i - 1])/(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-849de7332df28657","steps":["basis A013609","D+1"],"edges":["e-0b105283f866e3e7","e-963be6917ea50d69"],"text":"A088996 \u2192 basis A013609 \u2192 D+1"},"headline":"A088996 \u2192 basis A013609 \u2192 D+1","chain":null}