{"id":"n-3ce21439d055e53f","revision":0,"first_index":0,"fingerprint":"f6-635fb24c49f1a904","name":"one_plus_derivative of Triangle read by rows: T(n,k) is the number of induced subgraphs of the Lucas cube Lambda(n) that are isomorphic to the hypercube Q(k).","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"],["1"],["5","2"],["7","3"],["15","12","2"],["26","25","5"],["48","60","21","2"],["85","126","56","7"],["151","264","152","32","2"],["265","531","369","102","9"],["463","1050","865","310","45","2"],["804","2035","1936","847","165","11"],["1390","3888","4206","2192","555","60","2"],["2393","7332","8892","5382","1677","247","13"],["4105","13678","18396","12712","4739","910","77","2"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","7","10","29","56","131","274","601","1276","2735","5798","12293","25936","54619"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["5","1","-4"],["-7","-5","4"],["1","4","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["11","-1","-18","-8"],["-18","-15","10","8"],["4","14","8","0"],["3","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-10","-53","-91","-64","-16"],["-20","-56","-52","-16","0"],["-5","-9","-4","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171698,"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-a0d2964fda951ce7"]},{"id":171699,"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-a0d2964fda951ce7"]},{"id":171700,"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-a0d2964fda951ce7"]}],"region_exact":{},"incoming_edges":[{"id":"e-a0d2964fda951ce7","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-3ce21439d055e53f","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":"77fbe266b4034ba79a482c837881aeecd0c28c775a111fdc2b095f4a136256b9","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-299efe8218728fe3","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of induced subgraphs of the Lucas cube Lambda(n) that are isomorphic to the hypercube Q(k).","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:   1;\n  2:   5,   2;\n  3:   7,   3;\n  4:  15,  12,   2;\n  5:  26,  25,   5;\n  6:  48,  60,  21,   2;\n  7:  85, 126,  56,   7;\n  8: 151, 264, 152,  32, 2;\n  9: 265, 531, 369, 102, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^3+n^2-7*n+5+(4*n^2-5*n+1)*x+(4*n-4)*x^2)*P(n,x) = (3*n^3+4*n^2-18*n+11+(2*n^3+14*n^2-15*n-1)*x+(8*n^2+10*n-18)*x^2+(8*n-8)*x^3)*P(n-1,x) - (5*n^2+20*n+10+(9*n^2+56*n+53)*x+(4*n^2+52*n+91)*x^2+(16*n+64)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^3 + 4*n^2 - 18*n + 11 + (2*n^3 + 14*n^2 - 15*n - 1)*x + (8*n^2 + 10*n - 18)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - (5*n^2 + 20*n + 10 + (9*n^2 + 56*n + 53)*x + (4*n^2 + 52*n + 91)*x^2 + (16*n + 64)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 + n^2 - 7*n + 5 + (4*n^2 - 5*n + 1)*x + (4*n - 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((3*n^3 + 4*n^2 - 18*n + 11 + (2*n^3 + 14*n^2 - 15*n - 1)*x + (8*n^2 + 10*n - 18)*x^2 + (8*n - 8)*x^3)*v[i - 1] - (5*n^2 + 20*n + 10 + (9*n^2 + 56*n + 53)*x + (4*n^2 + 52*n + 91)*x^2 + (16*n + 64)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 + n^2 - 7*n + 5 + (4*n^2 - 5*n + 1)*x + (4*n - 4)*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-299efe8218728fe3","steps":["D+1"],"edges":["e-a0d2964fda951ce7"],"text":"A245962 \u2192 D+1"},"headline":"A245962 \u2192 D+1","chain":null}