{"id":"n-bad24ed0476f112e","revision":0,"first_index":0,"fingerprint":"f6-e4f1510ceee31d82","name":"derivative of reversal of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["3","2"],["6","10","3"],["10","30","21","4"],["15","70","84","36","5"],["21","140","252","180","55","6"],["28","252","630","660","330","78","7"],["36","420","1386","1980","1430","546","105","8"],["45","660","2772","5148","5005","2730","840","136","9"],["55","990","5148","12012","15015","10920","4760","1224","171","10"],["66","1430","9009","25740","40040","37128","21420","7752","1710","210","11"],["78","2002","15015","51480","97240","111384","81396","38760","11970","2310","253","12"],["91","2730","24024","97240","218790","302328","271320","162792","65835","17710","3036","300","13"],["105","3640","37128","175032","461890","755820","813960","596904","302841","106260","25300","3900","351","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","5","19","65","210","654","1985","5911","17345","50305","144516","411900","1166209","3283145"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["-1","0"],["1","1"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","4","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172434,"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-bb43596020d25079"]},{"id":172435,"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-bb43596020d25079"]},{"id":172436,"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-bb43596020d25079"]}],"region_exact":{},"incoming_edges":[{"id":"e-bb43596020d25079","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-bad24ed0476f112e","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":"30e251677f6e95522ae246ec761542690c9b5298518ab8b587cffe368b505c0d","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-4c232e962100ae75","role":"sequence","name":"reversal of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","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-225f52cb5b9a5a7e","kind":"fingerprint_related","node_a_id":"n-dc3e6727bddaa3d6","node_b_id":"n-bad24ed0476f112e","fingerprint":"f6-e4f1510ceee31d82","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e4f1510ceee31d82"},"other_id":"n-dc3e6727bddaa3d6","other":{"id":"n-dc3e6727bddaa3d6","name":"derivative of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","first_index":0,"fingerprint":"f6-e4f1510ceee31d82","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["1"]],"card_rows":[["1"],["3","2"],["6","10","3"],["10","30","21","4"]],"row_sums_preview":["0","0","1","5","19","65","210","654"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0eba401b117f16a9","steps":["D"],"edges":["e-04abbda3a865d9ca"],"text":"A085478 \u2192 D"},"headline":"A085478 \u2192 D","chain":null}},{"id":"l-49bcfecc5e93713f","kind":"fingerprint_related","node_a_id":"n-69a7d6f31c3b5900","node_b_id":"n-bad24ed0476f112e","fingerprint":"f6-e4f1510ceee31d82","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e4f1510ceee31d82"},"other_id":"n-69a7d6f31c3b5900","other":{"id":"n-69a7d6f31c3b5900","name":"derivative of Even row Pascal-square read by antidiagonals.","first_index":0,"fingerprint":"f6-e4f1510ceee31d82","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["4","6"]],"card_rows":[["2"],["4","6"],["6","20","12"],["8","42","60","20"]],"row_sums_preview":["0","2","10","38","130","420","1308","3970"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79ebafb266da3b41","steps":["D"],"edges":["e-032b739c94c6fe41"],"text":"A172431 \u2192 D"},"headline":"A172431 \u2192 D","chain":null}},{"id":"l-829b91981be5922b","kind":"fingerprint_related","node_a_id":"n-bad24ed0476f112e","node_b_id":"n-8bc9afb9e9e8548a","fingerprint":"f6-e4f1510ceee31d82","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e4f1510ceee31d82"},"other_id":"n-8bc9afb9e9e8548a","other":{"id":"n-8bc9afb9e9e8548a","name":"derivative of reversal of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","first_index":1,"fingerprint":"f6-e4f1510ceee31d82","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["4","6"]],"card_rows":[["2"],["4","6"],["6","20","12"],["8","42","60","20"]],"row_sums_preview":["0","2","10","38","130","420","1308","3970"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-976cdb6e4a693ec8","steps":["reverse","D"],"edges":["e-32f4efbf40925a37","e-018d442f6a54a658"],"text":"A078812 \u2192 reverse \u2192 D"},"headline":"A078812 \u2192 reverse \u2192 D","chain":null}},{"id":"l-acc3b6d845f9e525","kind":"fingerprint_related","node_a_id":"n-bad24ed0476f112e","node_b_id":"n-bd0eb2320338fabf","fingerprint":"f6-e4f1510ceee31d82","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e4f1510ceee31d82"},"other_id":"n-bd0eb2320338fabf","other":{"id":"n-bd0eb2320338fabf","name":"derivative of basis_map_A013611 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","first_index":0,"fingerprint":"f6-e4f1510ceee31d82","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["8"],["64","96"]],"card_rows":[["8"],["64","96"],["384","1280","768"],["2048","10752","15360","5120"]],"row_sums_preview":["0","8","160","2432","33280","430080","5357568","65044480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2245b35ec7cd4857","steps":["basis A013611","D"],"edges":["e-61d2480a3633cd97","e-5128f4e1d894b29c"],"text":"A091044 \u2192 basis A013611 \u2192 D"},"headline":"A091044 \u2192 basis A013611 \u2192 D","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:  1;\n  2:  3,   2;\n  3:  6,  10,    3;\n  4: 10,  30,   21,    4;\n  5: 15,  70,   84,   36,    5;\n  6: 21, 140,  252,  180,   55,    6;\n  7: 28, 252,  630,  660,  330,   78,   7;\n  8: 36, 420, 1386, 1980, 1430,  546, 105,   8;\n  9: 45, 660, 2772, 5148, 5005, 2730, 840, 136, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-3*n+1)*P(n,x) = (2*n^2+n-1+(n^2+n)*x)*P(n-1,x) + (4*n*x+n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n - 1 + (n^2 + n)*x)*P[n - 1, x] + (4*n*x + n*x^2)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + n - 1 + (n^2 + n)*x)*v[i - 1] + (4*n*x + n*x^2)*deriv(v[i - 1], x))/(2*n^2 - 3*n + 1)); 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-6758fe7729845be8","steps":["reverse","D"],"edges":["e-597fd348d5cdfa65","e-bb43596020d25079"],"text":"A054142 \u2192 reverse \u2192 D"},"headline":"A054142 \u2192 reverse \u2192 D","chain":null}
