{"id":"n-1e8b302e37f8b9ac","revision":0,"first_index":0,"fingerprint":"f6-b1032d52426bb000","name":"one_plus_derivative of derivative of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[[],[],["2"],["4"],["16","8"],["32","16"],["72","96","24"],["144","192","48"],["256","576","384","64"],["512","1152","768","128"],["800","2560","2880","1280","160"],["1600","5120","5760","2560","320"],["2304","9600","15360","11520","3840","384"],["4608","19200","30720","23040","7680","768"],["6272","32256","67200","71680","40320","10752","896"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","2","4","24","48","192","384","1280","2560","7680","15360","43008","86016","229376"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["4","20","-8"],["4","-18","4"],["-5","4","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["-64","160","-64","0"],["128","-166","54","-12"],["-75","59","-22","4"],["16","-10","4","0"],["-1","1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-16","32","16","-32","0"],["32","-20","-40","4","-8"],["-18","-8","2","-8","0"],["2","0","-2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172021,"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-a25ae42e5145f0dd"]},{"id":172022,"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-a25ae42e5145f0dd"]},{"id":172023,"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-a25ae42e5145f0dd"]}],"region_exact":{},"incoming_edges":[{"id":"e-a25ae42e5145f0dd","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-1e8b302e37f8b9ac","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":"d3c646e91eb838a172ce0eb70867246076aac1daf8e498c12bd4006fcb178261","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-507d474f4c2321e5","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","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-41849b1aa532612e","kind":"fingerprint_related","node_a_id":"n-9414feeb9954bdfa","node_b_id":"n-1e8b302e37f8b9ac","fingerprint":"f6-b1032d52426bb000","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1032d52426bb000"},"other_id":"n-9414feeb9954bdfa","other":{"id":"n-9414feeb9954bdfa","name":"theta_plus_one of basis_map_A013609 of Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-b1032d52426bb000","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["2","4"]],"card_rows":[["1"],["1"],["2","4"],["2","4"]],"row_sums_preview":["1","1","6","6","32","32","160","160"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4124f4242e7d4c67","steps":["basis A013609","\u03b8+1"],"edges":["e-500dd9b78517f374","e-8c490fa1c06d68a0"],"text":"A152198 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A152198 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-6a9f549cd1b08747","kind":"fingerprint_related","node_a_id":"n-78db6be9828bacac","node_b_id":"n-1e8b302e37f8b9ac","fingerprint":"f6-b1032d52426bb000","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1032d52426bb000"},"other_id":"n-78db6be9828bacac","other":{"id":"n-78db6be9828bacac","name":"one_plus_derivative of Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-b1032d52426bb000","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["2","1"]],"row_sums_preview":["1","1","3","3","8","8","20","20"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4124f4242e7d4c67","steps":["D+1"],"edges":["e-84a84c981d7d134c"],"text":"A152198 \u2192 D+1"},"headline":"A152198 \u2192 D+1","chain":null}},{"id":"l-b4f395def8ce850c","kind":"fingerprint_related","node_a_id":"n-39047ba7ddceb09b","node_b_id":"n-1e8b302e37f8b9ac","fingerprint":"f6-b1032d52426bb000","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1032d52426bb000"},"other_id":"n-39047ba7ddceb09b","other":{"id":"n-39047ba7ddceb09b","name":"theta_plus_one of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","first_index":0,"fingerprint":"f6-b1032d52426bb000","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["2","4"]],"card_rows":[["1"],["2"],["2","4"],["4","8"]],"row_sums_preview":["1","2","6","12","32","64","160","320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b6aca576a689e9f0","steps":["\u03b8+1"],"edges":["e-aaacb3fb62ab0797"],"text":"A274496 \u2192 \u03b8+1"},"headline":"A274496 \u2192 \u03b8+1","chain":null}},{"id":"l-c1aa991026cc809e","kind":"fingerprint_related","node_a_id":"n-b6f36b24716e535e","node_b_id":"n-1e8b302e37f8b9ac","fingerprint":"f6-b1032d52426bb000","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1032d52426bb000"},"other_id":"n-b6f36b24716e535e","other":{"id":"n-b6f36b24716e535e","name":"theta_plus_one of Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-b1032d52426bb000","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["1","2"]],"card_rows":[["1"],["1"],["1","2"],["1","2"]],"row_sums_preview":["1","1","3","3","8","8","20","20"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4124f4242e7d4c67","steps":["\u03b8+1"],"edges":["e-62712304e6278432"],"text":"A152198 \u2192 \u03b8+1"},"headline":"A152198 \u2192 \u03b8+1","chain":null}},{"id":"l-d7d45bb48add2322","kind":"fingerprint_related","node_a_id":"n-1e8b302e37f8b9ac","node_b_id":"n-3ea52383ec2ca4b4","fingerprint":"f6-b1032d52426bb000","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1032d52426bb000"},"other_id":"n-3ea52383ec2ca4b4","other":{"id":"n-3ea52383ec2ca4b4","name":"theta_plus_one of derivative of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","first_index":0,"fingerprint":"f6-b1032d52426bb000","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["4"],["8","16"],["16","32"]],"row_sums_preview":["0","0","2","4","24","48","192","384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b6aca576a689e9f0","steps":["D","\u03b8+1"],"edges":["e-5c70b541165ff56b","e-9eb147d30abdfc7c"],"text":"A274496 \u2192 D \u2192 \u03b8+1"},"headline":"A274496 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-dc4939b3bd729c0c","kind":"fingerprint_related","node_a_id":"n-4ecde6837aa04049","node_b_id":"n-1e8b302e37f8b9ac","fingerprint":"f6-b1032d52426bb000","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1032d52426bb000"},"other_id":"n-4ecde6837aa04049","other":{"id":"n-4ecde6837aa04049","name":"one_plus_derivative of basis_map_A013609 of Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-b1032d52426bb000","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["4","2"]],"card_rows":[["1"],["1"],["4","2"],["4","2"]],"row_sums_preview":["1","1","6","6","32","32","160","160"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4124f4242e7d4c67","steps":["basis A013609","D+1"],"edges":["e-500dd9b78517f374","e-eaf07c11792bcba9"],"text":"A152198 \u2192 basis A013609 \u2192 D+1"},"headline":"A152198 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-f689c13d70dc584b","kind":"fingerprint_related","node_a_id":"n-6e657ea7ab79a5f4","node_b_id":"n-1e8b302e37f8b9ac","fingerprint":"f6-b1032d52426bb000","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b1032d52426bb000"},"other_id":"n-6e657ea7ab79a5f4","other":{"id":"n-6e657ea7ab79a5f4","name":"one_plus_derivative of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","first_index":0,"fingerprint":"f6-b1032d52426bb000","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["4","2"]],"card_rows":[["1"],["2"],["4","2"],["8","4"]],"row_sums_preview":["1","2","6","12","32","64","160","320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b6aca576a689e9f0","steps":["D+1"],"edges":["e-240df8f0760fe339"],"text":"A274496 \u2192 D+1"},"headline":"A274496 \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:   0;\n  1:   0;\n  2:   2;\n  3:   4;\n  4:  16,    8;\n  5:  32,   16;\n  6:  72,   96,  24;\n  7: 144,  192,  48;\n  8: 256,  576, 384,  64;\n  9: 512, 1152, 768, 128;\n","recurrence_sentence":"The row polynomials satisfy (n^3-5*n^2+4*n+4+(4*n^2-18*n+20)*x+(4*n-8)*x^2)*P(n,x) = -(n^4-16*n^3+75*n^2-128*n+64-(n^4-10*n^3+59*n^2-166*n+160)*x-(4*n^3-22*n^2+54*n-64)*x^2-(4*n^2-12*n)*x^3)*P(n-1,x) + (2*n^3-18*n^2+32*n-16-(8*n^2+20*n-32)*x-(2*n^3-2*n^2+40*n-16)*x^2-(8*n^2-4*n+32)*x^3-8*n*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - 16*n^3 + 75*n^2 - 128*n + 64 - (n^4 - 10*n^3 + 59*n^2 - 166*n + 160)*x - (4*n^3 - 22*n^2 + 54*n - 64)*x^2 - (4*n^2 - 12*n)*x^3)*P[n - 1, x] + (2*n^3 - 18*n^2 + 32*n - 16 - (8*n^2 + 20*n - 32)*x - (2*n^3 - 2*n^2 + 40*n - 16)*x^2 - (8*n^2 - 4*n + 32)*x^3 - 8*n*x^4)*D[P[n - 1, x], x])/(n^3 - 5*n^2 + 4*n + 4 + (4*n^2 - 18*n + 20)*x + (4*n - 8)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = (-(n^4 - 16*n^3 + 75*n^2 - 128*n + 64 - (n^4 - 10*n^3 + 59*n^2 - 166*n + 160)*x - (4*n^3 - 22*n^2 + 54*n - 64)*x^2 - (4*n^2 - 12*n)*x^3)*v[i - 1] + (2*n^3 - 18*n^2 + 32*n - 16 - (8*n^2 + 20*n - 32)*x - (2*n^3 - 2*n^2 + 40*n - 16)*x^2 - (8*n^2 - 4*n + 32)*x^3 - 8*n*x^4)*deriv(v[i - 1], x))/(n^3 - 5*n^2 + 4*n + 4 + (4*n^2 - 18*n + 20)*x + (4*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-b6aca576a689e9f0","steps":["D","D+1"],"edges":["e-5c70b541165ff56b","e-a25ae42e5145f0dd"],"text":"A274496 \u2192 D \u2192 D+1"},"headline":"A274496 \u2192 D \u2192 D+1","chain":null}