{"id":"n-26ecae263b1ee6f4","revision":0,"first_index":0,"fingerprint":"f6-1149272fd8ce7a1a","name":"one_plus_derivative of Another version of Fibonacci-Pascal triangle A037027.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","1"],["1","5","2"],["0","4","11","3"],["0","2","16","25","5"],["0","0","9","43","50","8"],["0","0","3","37","109","98","13"],["0","0","0","16","114","250","185","21"],["0","0","0","4","71","320","548","343","34"],["0","0","0","0","25","245","817","1151","625","55"],["0","0","0","0","5","121","755","1969","2348","1125","89"],["0","0","0","0","0","36","461","2113","4522","4674","2004","144"],["0","0","0","0","0","6","190","1547","5536","10014","9126","3540","233"],["0","0","0","0","0","0","49","791","4697","13751","21520","17532","6209","377"],["0","0","0","0","0","0","7","281","2870","13265","32755","45124","33227","10825","610"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","8","18","48","110","260","586","1320","2918","6412","13954","30192","64926","138964"],"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","1"]],"recurrence":{"denominator":{"coeffs":[["2","3","-6","-6","5"],["-3","-9","-2","10","0"],["1","4","5","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","-2","-5","1","9","5"],["0","-1","-2","7","18","10","0"],["0","1","5","9","5","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-4","-7","-1","5"],["0","-2","-5","3","10","0"],["0","1","4","5","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171890,"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-041079b53a5d7e1b"]},{"id":171891,"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-041079b53a5d7e1b"]},{"id":171892,"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-041079b53a5d7e1b"]}],"region_exact":{},"incoming_edges":[{"id":"e-041079b53a5d7e1b","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-26ecae263b1ee6f4","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":"4f69587a6ec6f7cf58b75ff38ecc8cde30ce0911ee92068a97f2132af55c0f3b","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-e7c079f81f452bf3","role":"sequence","name":"Another version of Fibonacci-Pascal triangle A037027.","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, 1;\n  2: 1, 5,  2;\n  3: 0, 4, 11,  3;\n  4: 0, 2, 16, 25,   5;\n  5: 0, 0,  9, 43,  50,   8;\n  6: 0, 0,  3, 37, 109,  98,  13;\n  7: 0, 0,  0, 16, 114, 250, 185,   21;\n  8: 0, 0,  0,  4,  71, 320, 548,  343,  34;\n  9: 0, 0,  0,  0,  25, 245, 817, 1151, 625, 55;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2+(4*n^2-9*n+3)*x+(5*n^2-2*n-6)*x^2+(10*n-6)*x^3+5*x^4)*P(n,x) = ((n^2-n)*x+(5*n^2-2*n-2)*x^2+(9*n^2+7*n-5)*x^3+(5*n^2+18*n+1)*x^4+(10*n+9)*x^5+5*x^6)*P(n-2,x) + ((n^2-2*n-1)*x+(4*n^2-5*n-4)*x^2+(5*n^2+3*n-7)*x^3+(10*n-1)*x^4+5*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(((n^2 - n)*x + (5*n^2 - 2*n - 2)*x^2 + (9*n^2 + 7*n - 5)*x^3 + (5*n^2 + 18*n + 1)*x^4 + (10*n + 9)*x^5 + 5*x^6)*P[n - 2, x] + ((n^2 - 2*n - 1)*x + (4*n^2 - 5*n - 4)*x^2 + (5*n^2 + 3*n - 7)*x^3 + (10*n - 1)*x^4 + 5*x^5)*P[n - 1, x])/(n^2 - 3*n + 2 + (4*n^2 - 9*n + 3)*x + (5*n^2 - 2*n - 6)*x^2 + (10*n - 6)*x^3 + 5*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (((n^2 - n)*x + (5*n^2 - 2*n - 2)*x^2 + (9*n^2 + 7*n - 5)*x^3 + (5*n^2 + 18*n + 1)*x^4 + (10*n + 9)*x^5 + 5*x^6)*v[i - 2] + ((n^2 - 2*n - 1)*x + (4*n^2 - 5*n - 4)*x^2 + (5*n^2 + 3*n - 7)*x^3 + (10*n - 1)*x^4 + 5*x^5)*v[i - 1])/(n^2 - 3*n + 2 + (4*n^2 - 9*n + 3)*x + (5*n^2 - 2*n - 6)*x^2 + (10*n - 6)*x^3 + 5*x^4)); 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-e7c079f81f452bf3","steps":["D+1"],"edges":["e-041079b53a5d7e1b"],"text":"A111006 \u2192 D+1"},"headline":"A111006 \u2192 D+1","chain":null}