{"id":"n-c28e607bfa1726cb","revision":0,"first_index":0,"fingerprint":"f6-20fb9c15ea50c91a","name":"one_plus_derivative of T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["5","2"],["12","8"],["25","28","4"],["46","88","24"],["77","238","108","8"],["120","560","416","64"],["177","1176","1368","352","16"],["250","2256","3888","1600","160"],["341","4026","9768","6160","1040","32"],["452","6776","22176","20416","5440","384"],["585","10868","46332","59488","23920","2880","64"],["742","16744","90376","155584","90272","17024","896"],["925","24934","166452","371800","298480","84000","7616","128"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","7","20","57","158","431","1160","3089","8154","21367","55644","144137","371638","954335"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["5","-12","4"],["-3","4","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["3","-14","20","-8"],["2","0","-20","8"],["0","-4","8","0"],["1","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-12","24","-16"],["0","-2","12","-16","0"],["0","2","-4","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172584,"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-7b18b44150baeb8d"]},{"id":172585,"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-7b18b44150baeb8d"]},{"id":172586,"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-7b18b44150baeb8d"]}],"region_exact":{},"incoming_edges":[{"id":"e-7b18b44150baeb8d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-c28e607bfa1726cb","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":"bfe5bb5890f4e0d8775ccdc3caa51076f7b395e67cce4643e5248265ad5f3261","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-286e213a8791bd45","role":"sequence","name":"T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), 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:   1;\n  1:   2;\n  2:   5,    2;\n  3:  12,    8;\n  4:  25,   28,    4;\n  5:  46,   88,   24;\n  6:  77,  238,  108,    8;\n  7: 120,  560,  416,   64;\n  8: 177, 1176, 1368,  352,  16;\n  9: 250, 2256, 3888, 1600, 160;\n","recurrence_sentence":"The row polynomials satisfy (n^3-3*n^2+5*n+(4*n^2-12*n)*x+4*n*x^2)*P(n,x) = (n^3+2*n+3+(2*n^3-4*n^2-14)*x+(8*n^2-20*n+20)*x^2+(8*n-8)*x^3)*P(n-1,x) + ((2*n^2-2*n+2)*x-(4*n^2-12*n+12)*x^2-(16*n-24)*x^3-16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 + 2*n + 3 + (2*n^3 - 4*n^2 - 14)*x + (8*n^2 - 20*n + 20)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] + ((2*n^2 - 2*n + 2)*x - (4*n^2 - 12*n + 12)*x^2 - (16*n - 24)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 5*n + (4*n^2 - 12*n)*x + 4*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^3 + 2*n + 3 + (2*n^3 - 4*n^2 - 14)*x + (8*n^2 - 20*n + 20)*x^2 + (8*n - 8)*x^3)*v[i - 1] + ((2*n^2 - 2*n + 2)*x - (4*n^2 - 12*n + 12)*x^2 - (16*n - 24)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 5*n + (4*n^2 - 12*n)*x + 4*n*x^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-286e213a8791bd45","steps":["D+1"],"edges":["e-7b18b44150baeb8d"],"text":"A105070 \u2192 D+1"},"headline":"A105070 \u2192 D+1","chain":null}