{"id":"n-f91294e5f7553d54","revision":0,"first_index":0,"fingerprint":"f6-74619c273d5282fa","name":"one_plus_derivative of basis_map_A007318 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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","1"],["26","15","2"],["227","216","65","6"],["2463","3340","1637","338","24"],["31818","56645","39262","13113","2074","120"],["476004","1055831","955942","449343","114484","14724","720"],["8086005","21555576","24232314","14828148","5299909","1095076","118908","5040"],["153678285","479604672","646380330","489894404","227390589","65762028","11451068","1077264","40320"],["3230022240","11568563073","18219404472","16523978918","9482092668","3554772525","864941804","130319964","10824336","362880"],["74377757790","300992978583","543259474182","574940167058","394238041238","182460473835","57471236654","12084663644","1606094856","119492640","3628800"],["1861930845495","8407906762464","17127895593123","20750761156162","16585081471065","9161880105556","3558636064969","967623894642","179301265828","21330625176","1437798240","39916800"],["50343080856075","251089199634780","570265531338807","779090938273630","712191663556029","458161077241144","212216706943765","71093571392358","17020536567532","2821002708648","303897019392","18730500480","479001600"],["1462039279662150","7985552395848885","20017566761735574","30467978603148783","31384634822370200","23075083495224895","12437586961724734","4965082949314145","1463455552505158","313165419433276","46972421165064","4625157021696","262656570240","6227020800"],["45387551763678600","269529706656681615","739465967772261306","1241513026330363665","1423920107360151772","1178996486963874713","725844022630495578","336974106659515983","118282707016189928","31143179679962760","6027647194926752","825969157870704","74913113038464","3944953877760","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","43","514","7802","143032","3067048","75220976","2075278960","63585282880","2141554009280","78623825499520","3124616415034240","133624012702745600","6117989424431180800"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["-1","-2","-1"],["4","6","2"],["-4","-4","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","0"],["-2","-2"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187211,"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-f04a32ee62d3087d"]},{"id":187212,"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-f04a32ee62d3087d"]},{"id":187213,"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-f04a32ee62d3087d"]},{"id":187214,"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-f04a32ee62d3087d"]}],"region_exact":{},"incoming_edges":[{"id":"e-f04a32ee62d3087d","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-f91294e5f7553d54","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":"b73804c4f5fb8b06537221c36db29377361fe181aa1b9df9051f2c6f253e5e6f","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-6e1be0f1b7cdfa83","role":"sequence","name":"basis_map_A007318 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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,           1;\n  2:         26,          15,           2;\n  3:        227,         216,          65,           6;\n  4:       2463,        3340,        1637,         338,         24;\n  5:      31818,       56645,       39262,       13113,       2074,        120;\n  6:     476004,     1055831,      955942,      449343,     114484,      14724,       720;\n  7:    8086005,    21555576,    24232314,    14828148,    5299909,    1095076,    118908,      5040;\n  8:  153678285,   479604672,   646380330,   489894404,  227390589,   65762028,  11451068,   1077264,    40320;\n  9: 3230022240, 11568563073, 18219404472, 16523978918, 9482092668, 3554772525, 864941804, 130319964, 10824336, 362880;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(4*n^3-6*n^2+2*n)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(2*n^2-2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (4*n^3 - 6*n^2 + 2*n)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (2*n^2 - 2*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] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (4*n^3 - 6*n^2 + 2*n)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (2*n^2 - 2*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-7fd16a32c438d522","steps":["basis A007318","D+1"],"edges":["e-08400314c95111bf","e-f04a32ee62d3087d"],"text":"A059364 \u2192 basis A007318 \u2192 D+1"},"headline":"A059364 \u2192 basis A007318 \u2192 D+1","chain":null}