{"id":"n-bf0f00698b963e13","revision":0,"first_index":0,"fingerprint":"f6-b08b42757ceb90b6","name":"derivative of Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 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"],["14","2"],["83","166","3"],["432","3100","1296","4"],["2181","39796","59694","8724","5"],["10930","435774","1570644","871548","54650","6"],["54679","4398474","31769385","42359180","10996185","328074","7"],["273428","42407656","554168580","1448588888","923614300","127222968","1913996","8"],["1367177","397965160","8820869484","40312168056","50390210070","17641738968","1392878060","10937416","9"],["6835926","3674107546","132237233688","982291405512","2123631767380","1473437108268","308553545272","14696430184","61523334","10"],["34179675","33583763854","1903340152431","21859171566888","75494434377510","90593321253012","38253550242054","5075573739816","151126937343","341796750","11"],["170898424","305111295444","26629365520776","456058148590396","2382689548254960","4539859495134024","3335765367556944","912116297180792","79888096562328","1525556477220","1879882664","12"],["854492173","2761655954556","365147599023018","9076855039207980","68958175000542435","196776456942635256","229572533099741132","110333080000867896","20422923838217955","1217158663410060","15189107750058","10253906076","13"],["4272460922","24940011011350","4935161232004428","174409961029238116","1870206511317712190","7661592063418985634","13336392930762277768","10215456084558647512","3366371720371881942","436024902573095290","18095591184016236","149640066068100","55541991986","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","16","252","4832","110400","2943552","89905984","3098189824","118968134400","5038589957120","233364478009344","11734839037353984","636740292055748608","37083659566339391488"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"],["14","2"],["83","166","3"]],"recurrence":{"denominator":{"coeffs":[["-3"],["4"]]},"terms":[{"coeff":{"coeffs":[["0","0","-16"],["0","0","-64"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","-8","24"],["0","-32","320"],["0","0","-128"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","-16","16"],["0","0","-64","64"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-5","-2","99"],["-20","112","-300"],["0","-32","272"],["0","0","-64"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-32","36"],["0","-16","112","-96"],["0","0","-64","64"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-14","42"],["24","-80"],["0","32"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-12","12"],["0","16","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188317,"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-9c566549912970bc"]},{"id":188318,"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-9c566549912970bc"]},{"id":188319,"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-9c566549912970bc"]},{"id":188320,"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-9c566549912970bc"]}],"region_exact":{},"incoming_edges":[{"id":"e-9c566549912970bc","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-bf0f00698b963e13","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":"978a69635f72aeef65afda6d93aee4a306eb16d79cf782d1b8900a4fead0a38d","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-723ec2078be12613","role":"sequence","name":"Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, 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:       0;\n  1:       1;\n  2:      14,         2;\n  3:      83,       166,          3;\n  4:     432,      3100,       1296,           4;\n  5:    2181,     39796,      59694,        8724,           5;\n  6:   10930,    435774,    1570644,      871548,       54650,           6;\n  7:   54679,   4398474,   31769385,    42359180,    10996185,      328074,          7;\n  8:  273428,  42407656,  554168580,  1448588888,   923614300,   127222968,    1913996,        8;\n  9: 1367177, 397965160, 8820869484, 40312168056, 50390210070, 17641738968, 1392878060, 10937416, 9;\n","recurrence_sentence":"The row polynomials satisfy (4*n-3)*P(n,x) = -(64*n+16)*x^2*P(n-4,x) - ((32*n+8)*x+(128*n^2-320*n-24)*x^2)*P(n-3,x) - ((64*n+16)*x^2-(64*n+16)*x^3)*d/dx P(n-3,x) - (20*n+5+(32*n^2-112*n+2)*x+(64*n^3-272*n^2+300*n-99)*x^2)*P(n-2,x) - ((16*n+4)*x+(64*n^2-112*n+32)*x^2-(64*n^2-96*n+36)*x^3)*d/dx P(n-2,x) + (24*n-14+(32*n^2-80*n+42)*x)*P(n-1,x) + ((16*n-12)*x-(16*n-12)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 14 + 2*x, P(3,x) = 83 + 166*x + 3*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 14 + 2*x;\nP[3, x_] := 83 + 166*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(64*n + 16)*x^2*P[n - 4, x] - ((32*n + 8)*x + (128*n^2 - 320*n - 24)*x^2)*P[n - 3, x] - ((64*n + 16)*x^2 - (64*n + 16)*x^3)*D[P[n - 3, x], x] - (20*n + 5 + (32*n^2 - 112*n + 2)*x + (64*n^3 - 272*n^2 + 300*n - 99)*x^2)*P[n - 2, x] - ((16*n + 4)*x + (64*n^2 - 112*n + 32)*x^2 - (64*n^2 - 96*n + 36)*x^3)*D[P[n - 2, x], x] + (24*n - 14 + (32*n^2 - 80*n + 42)*x)*P[n - 1, x] + ((16*n - 12)*x - (16*n - 12)*x^2)*D[P[n - 1, x], x])/(4*n - 3)]];\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] = 14 + 2*x; v[4] = 83 + 166*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(64*n + 16)*x^2*v[i - 4] - ((32*n + 8)*x + (128*n^2 - 320*n - 24)*x^2)*v[i - 3] - ((64*n + 16)*x^2 - (64*n + 16)*x^3)*deriv(v[i - 3], x) - (20*n + 5 + (32*n^2 - 112*n + 2)*x + (64*n^3 - 272*n^2 + 300*n - 99)*x^2)*v[i - 2] - ((16*n + 4)*x + (64*n^2 - 112*n + 32)*x^2 - (64*n^2 - 96*n + 36)*x^3)*deriv(v[i - 2], x) + (24*n - 14 + (32*n^2 - 80*n + 42)*x)*v[i - 1] + ((16*n - 12)*x - (16*n - 12)*x^2)*deriv(v[i - 1], x))/(4*n - 3)); 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-723ec2078be12613","steps":["D"],"edges":["e-9c566549912970bc"],"text":"A144441 \u2192 D"},"headline":"A144441 \u2192 D","chain":null}