{"id":"n-a9089d8e5df0a6be","revision":0,"first_index":0,"fingerprint":"f6-40f5e5ec9eda3abc","name":"derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs of consecutive 0's (0<=k<=ceiling(n/2)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["5"],["17","4"],["49","32"],["129","156","12"],["321","600","132"],["769","2004","840","32"],["1793","6096","4056","448"],["4097","17356","16500","3520","80"],["9217","47048","59676","20480","1360"],["20481","122820","198096","98400","12720","192"],["45057","311232","616080","413376","86640","3840"],["98305","769980","1820508","1571584","480480","41664","448"],["212993","1867704","5163060","5530880","2301600","325248","10304"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","5","21","81","297","1053","3645","12393","41553","137781","452709","1476225","4782969","15411789"],"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":[["5","-2"],["-5","-4"],["0","6"]]},"terms":[{"coeff":{"coeffs":[["-7","14","-16"],["-9","8","-8"],["0","10","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-5","22","-16"],["0","2","14","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174303,"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-b3fe3e01e185e298"]},{"id":174304,"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-b3fe3e01e185e298"]},{"id":174305,"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-b3fe3e01e185e298"]}],"region_exact":{},"incoming_edges":[{"id":"e-b3fe3e01e185e298","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-a9089d8e5df0a6be","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":"027a9fca2a0d37ff2d92ecbc2e6e27143f5f290ce7de13449f9c21de0709bb67","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-6120ac587df8130e","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs of consecutive 0's (0<=k<=ceiling(n/2)).","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:    5;\n  3:   17,     4;\n  4:   49,    32;\n  5:  129,   156,    12;\n  6:  321,   600,   132;\n  7:  769,  2004,   840,   32;\n  8: 1793,  6096,  4056,  448;\n  9: 4097, 17356, 16500, 3520, 80;\n","recurrence_sentence":"The row polynomials satisfy -(5*n-5-(6*n^2-4*n-2)*x)*P(n,x) = -(9*n+7-(10*n^2+8*n+14)*x-(8*n^2-8*n-16)*x^2)*P(n-1,x) - (1-(2*n-5)*x-(14*n+22)*x^2+(16*n+16)*x^3)*d/dx 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[(-(9*n + 7 - (10*n^2 + 8*n + 14)*x - (8*n^2 - 8*n - 16)*x^2)*P[n - 1, x] - (1 - (2*n - 5)*x - (14*n + 22)*x^2 + (16*n + 16)*x^3)*D[P[n - 1, x], x])/(-5*n + 5 + (6*n^2 - 4*n - 2)*x)]];\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] = (-(9*n + 7 - (10*n^2 + 8*n + 14)*x - (8*n^2 - 8*n - 16)*x^2)*v[i - 1] - (1 - (2*n - 5)*x - (14*n + 22)*x^2 + (16*n + 16)*x^3)*deriv(v[i - 1], x))/(-5*n + 5 + (6*n^2 - 4*n - 2)*x)); 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-6120ac587df8130e","steps":["D"],"edges":["e-b3fe3e01e185e298"],"text":"A119808 \u2192 D"},"headline":"A119808 \u2192 D","chain":null}