{"id":"n-3ea52383ec2ca4b4","revision":0,"first_index":0,"fingerprint":"f6-b1032d52426bb000","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).","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"],["8","16"],["16","32"],["24","96","72"],["48","192","144"],["64","384","576","256"],["128","768","1152","512"],["160","1280","2880","2560","800"],["320","2560","5760","5120","1600"],["384","3840","11520","15360","9600","2304"],["768","7680","23040","30720","19200","4608"],["896","10752","40320","71680","67200","32256","6272"]],"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:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-16","4","-4"],["8","-10","8"],["0","4","-5"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["-128","128","-32","0"],["72","-210","78","0"],["-8","76","-73","3"],["0","-6","20","-4"],["0","0","-1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-32","16","32","-16","0"],["8","-52","-24","36","0"],["0","8","-18","-24","2"],["0","0","2","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172024,"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-9eb147d30abdfc7c"]},{"id":172025,"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-9eb147d30abdfc7c"]},{"id":172026,"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-9eb147d30abdfc7c"]}],"region_exact":{},"incoming_edges":[{"id":"e-9eb147d30abdfc7c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-3ea52383ec2ca4b4","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"ba33e5339f1137fcae20ebd13b301f50899ee29528e9f962e5873e2932a8b4e0","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-08fb5a235d22a4b7","kind":"fingerprint_related","node_a_id":"n-b6f36b24716e535e","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-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-4b93f1d0d31f5207","kind":"fingerprint_related","node_a_id":"n-9414feeb9954bdfa","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-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-74dcc2e3a244a605","kind":"fingerprint_related","node_a_id":"n-39047ba7ddceb09b","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-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-985f6e048631fbb6","kind":"fingerprint_related","node_a_id":"n-6e657ea7ab79a5f4","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-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}},{"id":"l-9c3266b866e8bd05","kind":"fingerprint_related","node_a_id":"n-4ecde6837aa04049","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-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-c7d20af6a2713f0a","kind":"fingerprint_related","node_a_id":"n-78db6be9828bacac","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-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-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-1e8b302e37f8b9ac","other":{"id":"n-1e8b302e37f8b9ac","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).","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"],["16","8"],["32","16"]],"row_sums_preview":["0","0","2","4","24","48","192","384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}}],"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:   8,  16;\n  5:  16,  32;\n  6:  24,  96,   72;\n  7:  48, 192,  144;\n  8:  64, 384,  576, 256;\n  9: 128, 768, 1152, 512;\n","recurrence_sentence":"The row polynomials satisfy (8*n-16+(4*n^2-10*n+4)*x+(n^3-5*n^2+8*n-4)*x^2)*P(n,x) = -(8*n^2-72*n+128+(6*n^3-76*n^2+210*n-128)*x+(n^4-20*n^3+73*n^2-78*n+32)*x^2-(n^4-4*n^3+3*n^2)*x^3)*P(n-1,x) + (8*n-32+(8*n^2-52*n+16)*x+(2*n^3-18*n^2-24*n+32)*x^2-(24*n^2-36*n+16)*x^3-(2*n^3-2*n^2)*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[(-(8*n^2 - 72*n + 128 + (6*n^3 - 76*n^2 + 210*n - 128)*x + (n^4 - 20*n^3 + 73*n^2 - 78*n + 32)*x^2 - (n^4 - 4*n^3 + 3*n^2)*x^3)*P[n - 1, x] + (8*n - 32 + (8*n^2 - 52*n + 16)*x + (2*n^3 - 18*n^2 - 24*n + 32)*x^2 - (24*n^2 - 36*n + 16)*x^3 - (2*n^3 - 2*n^2)*x^4)*D[P[n - 1, x], x])/(8*n - 16 + (4*n^2 - 10*n + 4)*x + (n^3 - 5*n^2 + 8*n - 4)*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] = (-(8*n^2 - 72*n + 128 + (6*n^3 - 76*n^2 + 210*n - 128)*x + (n^4 - 20*n^3 + 73*n^2 - 78*n + 32)*x^2 - (n^4 - 4*n^3 + 3*n^2)*x^3)*v[i - 1] + (8*n - 32 + (8*n^2 - 52*n + 16)*x + (2*n^3 - 18*n^2 - 24*n + 32)*x^2 - (24*n^2 - 36*n + 16)*x^3 - (2*n^3 - 2*n^2)*x^4)*deriv(v[i - 1], x))/(8*n - 16 + (4*n^2 - 10*n + 4)*x + (n^3 - 5*n^2 + 8*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-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}
