{"id":"n-582f0cc7fb344588","revision":1,"first_index":0,"fingerprint":"f6-9299f477a14cf70c","name":"one_plus_derivative of Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1.","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"],["2","1"],["4","5","1"],["5","12","7","1"],["7","28","29","10","1"],["8","41","68","45","12","1"],["10","69","165","165","75","15","1"],["11","88","264","375","270","99","17","1"],["13","128","490","924","915","492","142","20","1"],["14","153","676","1558","2034","1551","692","174","22","1"],["16","205","1085","3070","5050","4983","2995","1090","230","25","1"],["17","236","1385","4485","8805","10906","8663","4415","1415","270","27","1"],["19","300","2031","7710","18060","27192","26795","17418","7440","2040","339","30","1"],["20","337","2472","10371","27540","48510","57976","47471","26628","10110","2520","387","32","1"],["22","413","3409","16275","49791","102396","144914","142747","98315","47208","15582","3423","469","35","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","10","25","75","175","500","1125","3125","6875","18750","40625","109375","234375","625000"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"],["4","5","1"],["5","12","7","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-6","-11","-6","-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["2","6","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161260,"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-b1073907eb08b9b5"]},{"id":161261,"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-b1073907eb08b9b5"]},{"id":161262,"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-b1073907eb08b9b5"]}],"region_exact":{},"incoming_edges":[{"id":"e-b1073907eb08b9b5","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-582f0cc7fb344588","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":"9bc5f008ac4ecec0b42f72340186637c8d696b1bd802604b2158b3c48ab3ccd2","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-4e0b19699f4cf0db","role":"sequence","name":"Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= 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":[{"id":"l-e11b33aa5a3fc35f","kind":"oeis_match","node_a_id":"n-582f0cc7fb344588","node_b_id":null,"fingerprint":null,"oeis_id":"A026950","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A026950","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A026950","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-582f0cc7fb344588","oeis_id":"A026950","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":31,"oeis_terms_available":31,"term_count":200,"raw_size":8261}],"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,   1;\n  2:  4,   5,   1;\n  3:  5,  12,   7,    1;\n  4:  7,  28,  29,   10,    1;\n  5:  8,  41,  68,   45,   12,    1;\n  6: 10,  69, 165,  165,   75,   15,   1;\n  7: 11,  88, 264,  375,  270,   99,  17,   1;\n  8: 13, 128, 490,  924,  915,  492, 142,  20,  1;\n  9: 14, 153, 676, 1558, 2034, 1551, 692, 174, 22, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+6*x+11*x^2+6*x^3+x^4)*P(n-4,x) + (2+6*x+2*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 4 + 5*x + x^2, P(3,x) = 5 + 12*x + 7*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 4 + 5*x + x^2;\nP[3, x_] := 5 + 12*x + 7*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[-(1 + 6*x + 11*x^2 + 6*x^3 + x^4)*P[n - 4, x] + (2 + 6*x + 2*x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 4 + 5*x + x^2; v[4] = 5 + 12*x + 7*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = -(1 + 6*x + 11*x^2 + 6*x^3 + x^4)*v[i - 4] + (2 + 6*x + 2*x^2)*v[i - 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-4e0b19699f4cf0db","steps":["D+1"],"edges":["e-b1073907eb08b9b5"],"text":"A026374 \u2192 D+1"},"headline":"A026374 \u2192 D+1","chain":null}
