{"id":"n-3ebe5425d87cb4dd","revision":0,"first_index":0,"fingerprint":"f6-fc2bebe71383f67b","name":"derivative of gamma_polynomial of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["2"],["8"],["20","48"],["40","368"],["70","1568","2160"],["112","4928","25344"],["168","12768","157056","161280"],["240","28896","689280","2594304"],["330","59136","2412960","21441024","18144000"],["440","111936","7180800","122535424","374860800"],["572","199056","18889728","547216384","3912629760","2874009600"],["728","336336","45082752","2040822784","27807037440","73047951360"],["910","544544","99425040","6624730112","152298764800","927566290944","610248038400"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","2","8","68","408","3798","30384","331272","3312720","42057450","504689400","7352945100","102941231400","1696837794750"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"],["8"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","-12"],["0","0","28"],["0","0","-23"],["0","0","8"],["0","0","-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","-4"],["0","6"],["0","-2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["-1","4"],["0","-6"],["0","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2"],["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177278,"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-dd8eab61ded1ae60"]},{"id":177279,"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-dd8eab61ded1ae60"]},{"id":177280,"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-dd8eab61ded1ae60"]}],"region_exact":{},"incoming_edges":[{"id":"e-dd8eab61ded1ae60","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-3ebe5425d87cb4dd","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":"36ef0b9d210c75cc1d62b83b053ac170f81131e145436775cdf7dccd9625b760","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-eea7ec082f5ce396","role":"sequence","name":"gamma_polynomial of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.","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:   0;\n  2:   2;\n  3:   8;\n  4:  20,    48;\n  5:  40,   368;\n  6:  70,  1568,   2160;\n  7: 112,  4928,  25344;\n  8: 168, 12768, 157056,  161280;\n  9: 240, 28896, 689280, 2594304;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(n^5-8*n^4+23*n^3-28*n^2+12*n)*x^2*P(n-4,x) - (2*n^3-6*n^2+4*n)*x*P(n-3,x) - (n-(2*n^3-6*n^2+4*n)*x)*P(n-2,x) + (2*n-2)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2, P(3,x) = 8.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[3, x_] := 8;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 8*n^4 + 23*n^3 - 28*n^2 + 12*n)*x^2*P[n - 4, x] - (2*n^3 - 6*n^2 + 4*n)*x*P[n - 3, x] - (n - (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 2, x] + (2*n - 2)*P[n - 1, x])/(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] = 0; v[3] = 2; v[4] = 8; for(n = 4, N, my(i = n + 1); v[i] = (-(n^5 - 8*n^4 + 23*n^3 - 28*n^2 + 12*n)*x^2*v[i - 4] - (2*n^3 - 6*n^2 + 4*n)*x*v[i - 3] - (n - (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 2] + (2*n - 2)*v[i - 1])/(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-08ac2c6d9914754c","steps":["\u03b3","D"],"edges":["e-36556341d5271807","e-dd8eab61ded1ae60"],"text":"A154986 \u2192 \u03b3 \u2192 D"},"headline":"A154986 \u2192 \u03b3 \u2192 D","chain":null}