{"id":"n-08469142fb9c5873","revision":1,"first_index":0,"fingerprint":"f6-4f4ca2dec4972ed6","name":"one_plus_derivative of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","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":true,"rows":[["1"],["1"],["3","2"],["3","2"],["5","12","4"],["5","12","4"],["7","30","36","8"],["7","30","36","8"],["9","56","120","96","16"],["9","56","120","96","16"],["11","90","280","400","240","32"],["11","90","280","400","240","32"],["13","132","540","1120","1200","576","64"],["13","132","540","1120","1200","576","64"],["15","182","924","2520","3920","3360","1344","128"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","5","5","21","21","81","81","297","297","1053","1053","3645","3645","12393"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2","-6","4"],["-1","4","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["2","-6","8","-8"],["-7","6","-12","8"],["3","-4","8","0"],["0","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-12","-16","-16"],["-2","-6","-12","-16","0"],["0","-2","-4","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161008,"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-90e13dda23ca7fe5"]},{"id":161009,"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-90e13dda23ca7fe5"]},{"id":161010,"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-90e13dda23ca7fe5"]}],"region_exact":{},"incoming_edges":[{"id":"e-90e13dda23ca7fe5","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-08469142fb9c5873","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":"b06f38d395a195a4c6edc90918862f4ddb71d871694016424a1900b66b352d6f","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-ba3bb18490b15e2e","role":"sequence","name":"Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","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-14efd2944a1e73f7","kind":"fingerprint_related","node_a_id":"n-08469142fb9c5873","node_b_id":"n-6dbf222e2d76f379","fingerprint":"f6-4f4ca2dec4972ed6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f4ca2dec4972ed6"},"other_id":"n-6dbf222e2d76f379","other":{"id":"n-6dbf222e2d76f379","name":"one_plus_derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","first_index":0,"fingerprint":"f6-4f4ca2dec4972ed6","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3"],["9","6"]],"card_rows":[["1"],["3"],["9","6"],["27","18"]],"row_sums_preview":["1","3","15","45","189","567","2187","6561"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c641d81e21514ce","steps":["D+1"],"edges":["e-dfefcaa0f678161b"],"text":"A274498 \u2192 D+1"},"headline":"A274498 \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: 1;\n  2: 3,  2;\n  3: 3,  2;\n  4: 5, 12,   4;\n  5: 5, 12,   4;\n  6: 7, 30,  36,  8;\n  7: 7, 30,  36,  8;\n  8: 9, 56, 120, 96, 16;\n  9: 9, 56, 120, 96, 16;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n-2+(4*n-6)*x+4*x^2)*P(n,x) = (3*n^2-7*n+2+(2*n^3-4*n^2+6*n-6)*x+(8*n^2-12*n+8)*x^2+(8*n-8)*x^3)*P(n-1,x) - (2*n+(2*n^2+6*n+4)*x+(4*n^2+12*n+12)*x^2+(16*n+16)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 7*n + 2 + (2*n^3 - 4*n^2 + 6*n - 6)*x + (8*n^2 - 12*n + 8)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - (2*n + (2*n^2 + 6*n + 4)*x + (4*n^2 + 12*n + 12)*x^2 + (16*n + 16)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^2 - n - 2 + (4*n - 6)*x + 4*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^2 - 7*n + 2 + (2*n^3 - 4*n^2 + 6*n - 6)*x + (8*n^2 - 12*n + 8)*x^2 + (8*n - 8)*x^3)*v[i - 1] - (2*n + (2*n^2 + 6*n + 4)*x + (4*n^2 + 12*n + 12)*x^2 + (16*n + 16)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^2 - n - 2 + (4*n - 6)*x + 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-ba3bb18490b15e2e","steps":["D+1"],"edges":["e-90e13dda23ca7fe5"],"text":"A188440 \u2192 D+1"},"headline":"A188440 \u2192 D+1","chain":null}