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T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","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":[[],["1"],["8","4"],["60","68","18"],["484","884","516","96"],["4283","10960","10224","4112","600"],["41604","138192","179916","114592","35640","4320"],["441840","1815224","3060036","2706432","1322840","338256","35280"],["5103716","25083488","52203132","59590112","40257080","16077504","3509856","322560"],["63784717","365970032","909801504","1279222384","1111893800","611290752","207386928","39649536","3265920"],["858229228","5642251236","16354965396","27421765216","29292748320","20662137120","9616858720","2845608192","485248320","36288000"],["12375808436","91864339052","304801894038","595076551232","756743106080","654660769632","389975869472","157850909440","41519363616","6402556800","439084800"],["190475376312","1577477897956","5904429654072","13180480256640","19491259005280","20041689365280","14615296088736","7555307619584","2711773119744","643191022080","90660936960","5748019200"],["3117335380103","28522663914400","119022461927184","299455574635616","505794946143520","604034320191936","522805087501648","330313305274112","151131451368960","48811518948480","10557106585344","1371957212160","80951270400"],["54069030540024","542070535375328","2497430903383272","6999861336553984","13313700435414320","18141862528916832","18216160000405920","13642338976792576","7617826006850880","3131274800115200","920520436235136","183181377042432","22103238044160","1220496076800"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","12","146","1980","30179","514264","9719908","202147448","4592265573","113216099748","3011710252598","86007788361844","2625018680353863","85283620101746864"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"],["8","4"],["60","68","18"]],"recurrence":{"denominator":{"coeffs":[["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0"],["1"],["-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["-2","4"],["-2","-6"],["2","2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-2","0","0"],["6","-4","2"],["1","2","-5"],["0","4","4"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","0"],["-6","4"],["-2","-6"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185570,"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-33272e50f5a1a631"]},{"id":185571,"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-33272e50f5a1a631"]},{"id":185572,"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-33272e50f5a1a631"]},{"id":185573,"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-33272e50f5a1a631"]}],"region_exact":{},"incoming_edges":[{"id":"e-33272e50f5a1a631","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-5433b0a58dea9df1","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":"72c168137f31cb9123374ab8a4ec1939b66bdf200128fd9b79cd29ede5b32f96","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-4a57e8264fe984f4","role":"sequence","name":"basis_map_A007318 of Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","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:        1;\n  2:        8,         4;\n  3:       60,        68,        18;\n  4:      484,       884,       516,         96;\n  5:     4283,     10960,     10224,       4112,        600;\n  6:    41604,    138192,    179916,     114592,      35640,      4320;\n  7:   441840,   1815224,   3060036,    2706432,    1322840,    338256,     35280;\n  8:  5103716,  25083488,  52203132,   59590112,   40257080,  16077504,   3509856,   322560;\n  9: 63784717, 365970032, 909801504, 1279222384, 1111893800, 611290752, 207386928, 39649536, 3265920;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (2*n^3-2*n^2-2*n+(2*n^3-6*n^2+4*n)*x)*P(n-3,x) - (n^4-n^2-6*n+2+(2*n^4-4*n^3-2*n^2+4*n)*x+(n^4-4*n^3+5*n^2-2*n)*x^2)*P(n-2,x) + (2*n^3-2*n^2-6*n+4+(2*n^3-6*n^2+4*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 8 + 4*x, P(3,x) = 60 + 68*x + 18*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 8 + 4*x;\nP[3, x_] := 60 + 68*x + 18*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (2*n^3 - 2*n^2 - 2*n + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 3, x] - (n^4 - n^2 - 6*n + 2 + (2*n^4 - 4*n^3 - 2*n^2 + 4*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*P[n - 2, x] + (2*n^3 - 2*n^2 - 6*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 1, x])/(n^2 - 3*n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 8 + 4*x; v[4] = 60 + 68*x + 18*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (2*n^3 - 2*n^2 - 2*n + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 3] - (n^4 - n^2 - 6*n + 2 + (2*n^4 - 4*n^3 - 2*n^2 + 4*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*v[i - 2] + (2*n^3 - 2*n^2 - 6*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 1])/(n^2 - 3*n + 2)); 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-f3e0631eb8740f76","steps":["basis A007318","D"],"edges":["e-3f96abc359582f71","e-33272e50f5a1a631"],"text":"A358735 \u2192 basis A007318 \u2192 D"},"headline":"A358735 \u2192 basis A007318 \u2192 D","chain":null}