{"id":"n-c1338451c8a75e40","revision":0,"first_index":0,"fingerprint":"f6-15d50c9a2b2cc2b4","name":"derivative of Triangle T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 1.","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"],["6","2"],["27","42","3"],["112","540","174","4"],["453","5756","5214","564","5"],["1818","56334","117960","34956","1590","6"],["7279","528822","2309865","1503020","188325","4110","7"],["29124","4861304","41717238","52974600","14294800","879360","10038","8"],["116505","44217720","718519500","1658102904","845704050","111703680","3720192","23592","9"],["466030","400056570","12026924640","48158754600","42881117580","10547704620","759376128","14665896","53982","10"],["1864131","3609829730","197831557935","1332256061160","1965109335630","825581572956","110128349142","4659371304","54895599","121170","11"],["7456536","32529478452","3219452372910","35680380224220","84066496694280","56960415494136","12827285355780","1006713927864","26456730768","197646900","268158","12"],["29826157","292944262932","52047974361018","934935537244300","3427798158516555","3598098727477752","1286122989305364","169502356310904","8308438921035","141500744980","690849522","587100","13"],["119304642","2637273846470","838040586509064","24136758721735764","135087427922149730","213327057758388642","115894126105335648","24018674775626136","1974984797202102","63279584148330","722085609080","2360181972","1274598","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","8","72","830","11992","212664","4541428","114766472","3382108152","114789120056","4439232958768","193819935650016","9477249348407632","515343712091312192"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["-4","4","-2"],["0","-2","3"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0"],["0","-5","5","-1"],["0","0","-4","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0"],["0","0","-1","1"],["0","0","0","-1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","2"],["5","-4"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-5","1"],["0","5","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"],["0","0","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184115,"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-a504ee9286a25f70"]},{"id":184116,"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-a504ee9286a25f70"]},{"id":184117,"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-a504ee9286a25f70"]}],"region_exact":{},"incoming_edges":[{"id":"e-a504ee9286a25f70","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-c1338451c8a75e40","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":"8fc690a4f54d67704ea7affbfc6c58048befe1e03c57f7c5c208332df5dd097b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0c97d42b0cd05e48","role":"sequence","name":"Triangle T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 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:      0;\n  1:      1;\n  2:      6,        2;\n  3:     27,       42,         3;\n  4:    112,      540,       174,          4;\n  5:    453,     5756,      5214,        564,         5;\n  6:   1818,    56334,    117960,      34956,      1590,         6;\n  7:   7279,   528822,   2309865,    1503020,    188325,      4110,       7;\n  8:  29124,  4861304,  41717238,   52974600,  14294800,    879360,   10038,     8;\n  9: 116505, 44217720, 718519500, 1658102904, 845704050, 111703680, 3720192, 23592, 9;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n+(2*n^2-4*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) - (5*n*x+(4*n^2-5*n)*x^2-(n^2-n)*x^3)*d/dx P(n-2,x) - (n*x^2+(n^2-n)*x^3)*d^2/dx^2 P(n-2,x) + (5*n-4+(2*n^2-4*n+2)*x)*P(n-1,x) + ((5*n-5)*x-(n-1)*x^2)*d/dx P(n-1,x) + (n-1)*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] - (5*n*x + (4*n^2 - 5*n)*x^2 - (n^2 - n)*x^3)*D[P[n - 2, x], x] - (n*x^2 + (n^2 - n)*x^3)*D[P[n - 2, x], {x, 2}] + (5*n - 4 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] + ((5*n - 5)*x - (n - 1)*x^2)*D[P[n - 1, x], x] + (n - 1)*x^2*D[P[n - 1, x], {x, 2}])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] - (5*n*x + (4*n^2 - 5*n)*x^2 - (n^2 - n)*x^3)*deriv(v[i - 2], x) - (n*x^2 + (n^2 - n)*x^3)*deriv(deriv(v[i - 2], x), x) + (5*n - 4 + (2*n^2 - 4*n + 2)*x)*v[i - 1] + ((5*n - 5)*x - (n - 1)*x^2)*deriv(v[i - 1], x) + (n - 1)*x^2*deriv(deriv(v[i - 1], x), x))/(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-0c97d42b0cd05e48","steps":["D"],"edges":["e-a504ee9286a25f70"],"text":"A166960 \u2192 D"},"headline":"A166960 \u2192 D","chain":null}