{"id":"n-bdfaac9060ae9acf","revision":0,"first_index":0,"fingerprint":"f6-0ea939b55f107842","name":"derivative of shift_x_plus_one of Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(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":false,"rows":[[],[],["2"],["6"],["24","12"],["80","60"],["270","300","60"],["882","1260","420"],["2856","5040","2520","280"],["9144","19152","12600","2520"],["29070","70560","57960","17640","1260"],["91850","253440","249480","101640","13860"],["288816","892980","1025640","526680","110880","5544"],["904488","3097380","4066920","2522520","720720","72072"],["2823002","10606596","15675660","11411400","4144140","648648","24024"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","2","6","36","140","630","2562","10696","43416","176490","710270","2850540","11384100","45333470"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-12","-12"],["5","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","-14","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174106,"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-563ad78586b3b53f"]},{"id":174107,"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-563ad78586b3b53f"]},{"id":174108,"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-563ad78586b3b53f"]}],"region_exact":{},"incoming_edges":[{"id":"e-563ad78586b3b53f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-bdfaac9060ae9acf","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":"d8cbfb35fca2a82fc54dcd3df76748732e83b16f57efbc417aed8ea68f2d7892","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-1e7342ec7a250847","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(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-26c2d30104d3f77c","kind":"fingerprint_related","node_a_id":"n-4ee9a668e5c066a0","node_b_id":"n-bdfaac9060ae9acf","fingerprint":"f6-0ea939b55f107842","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0ea939b55f107842"},"other_id":"n-4ee9a668e5c066a0","other":{"id":"n-4ee9a668e5c066a0","name":"derivative of shift_x_plus_one of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.","first_index":1,"fingerprint":"f6-0ea939b55f107842","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["6"],["24","12"],["80","60"]],"row_sums_preview":["0","0","2","6","36","140","630","2562"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4b168caf0bc63f8b","steps":["x \u21a6 x+1","D"],"edges":["e-502ba5da9494e9e3","e-26c849d5649379dd"],"text":"A089627 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A089627 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-73c79bb924f05721","kind":"fingerprint_related","node_a_id":"n-188454bd1b8bfd31","node_b_id":"n-bdfaac9060ae9acf","fingerprint":"f6-0ea939b55f107842","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0ea939b55f107842"},"other_id":"n-188454bd1b8bfd31","other":{"id":"n-188454bd1b8bfd31","name":"mobius_x_over_x_plus_one of A Motzkin related triangle.","first_index":0,"fingerprint":"f6-0ea939b55f107842","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","1","2"]],"card_rows":[["1"],["0","1"],["0","1","2"],["0","0","3","4"]],"row_sums_preview":["1","1","3","7","21","61","191","603"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-de4284a2d5d7d0a6","steps":["x \u21a6 x/(x+1)"],"edges":["e-bae25cc5f8e98b16"],"text":"A107131 \u2192 x \u21a6 x/(x+1)"},"headline":"A107131 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7d763845a20d529d","kind":"fingerprint_related","node_a_id":"n-3e465972400fd143","node_b_id":"n-bdfaac9060ae9acf","fingerprint":"f6-0ea939b55f107842","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0ea939b55f107842"},"other_id":"n-3e465972400fd143","other":{"id":"n-3e465972400fd143","name":"shift_x_plus_one of Triangular array of Motzkin polynomial coefficients.","first_index":0,"fingerprint":"f6-0ea939b55f107842","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["4","3"]],"row_sums_preview":["1","1","3","7","21","61","191","603"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-169bd483754ab390","steps":["basis A007318"],"edges":["e-5fd319f73e442e4d"],"text":"A055151 \u2192 basis A007318"},"headline":"A055151 \u2192 basis A007318","chain":null}},{"id":"l-99cdf65dfc9e2715","kind":"fingerprint_related","node_a_id":"n-6288e428535e0ce0","node_b_id":"n-bdfaac9060ae9acf","fingerprint":"f6-0ea939b55f107842","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0ea939b55f107842"},"other_id":"n-6288e428535e0ce0","other":{"id":"n-6288e428535e0ce0","name":"shift_x_plus_one of gamma_polynomial of narayana","first_index":1,"fingerprint":"f6-0ea939b55f107842","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["4","3"]],"row_sums_preview":["1","1","3","7","21","61","191","603"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["\u03b3","basis A007318"],"edges":["e-dbb6454df235fcfe","e-4e65d47143297ee9"],"text":"narayana \u2192 \u03b3 \u2192 basis A007318"},"headline":"narayana \u2192 \u03b3 \u2192 basis A007318","chain":null}},{"id":"l-aaa29fb4276a68c3","kind":"fingerprint_related","node_a_id":"n-5af2cb3fcde9e652","node_b_id":"n-bdfaac9060ae9acf","fingerprint":"f6-0ea939b55f107842","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0ea939b55f107842"},"other_id":"n-5af2cb3fcde9e652","other":{"id":"n-5af2cb3fcde9e652","name":"shift_x_plus_one of gamma_polynomial of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","first_index":0,"fingerprint":"f6-0ea939b55f107842","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["6","3"]],"card_rows":[["1"],["2"],["6","3"],["16","12"]],"row_sums_preview":["1","2","9","28","105","366","1337","4824"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-984320118b653f6b","steps":["\u03b3","x \u21a6 x+1"],"edges":["e-cc6de87a980d771e","e-6d7e509e8f17ee7b"],"text":"A132812 \u2192 \u03b3 \u2192 x \u21a6 x+1"},"headline":"A132812 \u2192 \u03b3 \u2192 x \u21a6 x+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:    6;\n  4:   24,    12;\n  5:   80,    60;\n  6:  270,   300,    60;\n  7:  882,  1260,   420;\n  8: 2856,  5040,  2520,  280;\n  9: 9144, 19152, 12600, 2520;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (5*n-12+(4*n-12)*x)*P(n-1,x) - (6+14*x+8*x^2)*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[((5*n - 12 + (4*n - 12)*x)*P[n - 1, x] - (6 + 14*x + 8*x^2)*D[P[n - 1, x], x])/(n - 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] = ((5*n - 12 + (4*n - 12)*x)*v[i - 1] - (6 + 14*x + 8*x^2)*deriv(v[i - 1], x))/(n - 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-8bfdd58279be42d7","steps":["x \u21a6 x+1","D"],"edges":["e-0f1b9062159c7e91","e-563ad78586b3b53f"],"text":"A391403 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A391403 \u2192 x \u21a6 x+1 \u2192 D","chain":null}
