{"id":"n-2ce402adf26dff78","revision":0,"first_index":0,"fingerprint":"f6-ef2fcf41caa3ea75","name":"one_plus_derivative of veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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","1"],["14","6"],["40","26","1"],["112","100","10"],["304","360","63","1"],["800","1232","322","14"],["2048","4032","1456","116","1"],["5120","12672","6048","744","18"],["12544","38400","23520","4080","185","1"],["30208","112640","86592","20064","1430","22"],["71680","321024","304128","90816","9240","270","1"],["167936","891904","1025024","384384","52624","2444","26"],["389120","2422784","3331328","1537536","272272","18200","371","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","6","20","67","222","728","2368","7653","24602","78730","250956","797159","2524342","7971612"],"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"]],"recurrence":{"denominator":{"coeffs":[["-20","52","-8"],["26","-60","8"],["-8","8","0"],["2","0","0"]]},"terms":[{"coeff":{"coeffs":[["-32","112","6","-4"],["36","-125","-10","4"],["-8","12","4","0"],["4","1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-64","-128","68","-8"],["0","-8","34","-8","0"],["0","8","-2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173852,"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-db50ed911a521fb6"]},{"id":173853,"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-db50ed911a521fb6"]},{"id":173854,"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-db50ed911a521fb6"]}],"region_exact":{},"incoming_edges":[{"id":"e-db50ed911a521fb6","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-2ce402adf26dff78","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":"8e7472fa1b7c5477b33ce1738ea59de27c96d5f57529fffc1c4d7e71f063b050","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-b6912a345c4fb865","role":"sequence","name":"veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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-7b244d7fec87eb12","kind":"fingerprint_related","node_a_id":"n-2ce402adf26dff78","node_b_id":"n-7f993dd5772dfd1e","fingerprint":"f6-ef2fcf41caa3ea75","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ef2fcf41caa3ea75"},"other_id":"n-7f993dd5772dfd1e","other":{"id":"n-7f993dd5772dfd1e","name":"one_plus_derivative of veronese_section_2_0 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-ef2fcf41caa3ea75","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3"],["6"],["15","3"]],"card_rows":[["3"],["6"],["15","3"],["42","18"]],"row_sums_preview":["3","6","18","60","201","666","2184","7104"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["Veronese 2","D+1"],"edges":["e-74cdb64b6d1e6b98","e-fbbefc9b0fe82331"],"text":"A120910 \u2192 Veronese 2 \u2192 D+1"},"headline":"A120910 \u2192 Veronese 2 \u2192 D+1","chain":null}}],"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,     1;\n  3:   14,     6;\n  4:   40,    26,    1;\n  5:  112,   100,   10;\n  6:  304,   360,   63,   1;\n  7:  800,  1232,  322,  14;\n  8: 2048,  4032, 1456, 116,  1;\n  9: 5120, 12672, 6048, 744, 18;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-8*n^2+26*n-20+(8*n^2-60*n+52)*x+(8*n-8)*x^2)*P(n,x) = (4*n^3-8*n^2+36*n-32+(n^3+12*n^2-125*n+112)*x+(4*n^2-10*n+6)*x^2+(4*n-4)*x^3)*P(n-1,x) + ((8*n^2-8*n-64)*x-(2*n^2-34*n+128)*x^2-(8*n-68)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 8*n^2 + 36*n - 32 + (n^3 + 12*n^2 - 125*n + 112)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + ((8*n^2 - 8*n - 64)*x - (2*n^2 - 34*n + 128)*x^2 - (8*n - 68)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - 8*n^2 + 26*n - 20 + (8*n^2 - 60*n + 52)*x + (8*n - 8)*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^3 - 8*n^2 + 36*n - 32 + (n^3 + 12*n^2 - 125*n + 112)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)*v[i - 1] + ((8*n^2 - 8*n - 64)*x - (2*n^2 - 34*n + 128)*x^2 - (8*n - 68)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - 8*n^2 + 26*n - 20 + (8*n^2 - 60*n + 52)*x + (8*n - 8)*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-b6912a345c4fb865","steps":["D+1"],"edges":["e-db50ed911a521fb6"],"text":"A372868 \u2192 D+1"},"headline":"A372868 \u2192 D+1","chain":null}
