{"id":"n-26e4387737b6859f","revision":0,"first_index":1,"fingerprint":"f6-285a8f351a67f82b","name":"one_plus_derivative of reversal of Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","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"],["2","1"],["4","5","1"],["6","17","9","1"],["8","37","45","14","1"],["10","65","133","95","20","1"],["12","101","297","364","175","27","1"],["14","145","561","1005","840","294","35","1"],["16","197","949","2266","2805","1722","462","44","1"],["18","257","1485","4459","7436","6831","3234","690","54","1"],["20","325","2193","7960","16835","21021","15015","5676","990","65","1"],["22","401","3097","13209","34000","54236","53053","30459","9438","1375","77","1"],["24","485","4221","20710","62985","123012","154700","122408","57915","15015","1859","90","1"],["26","577","5589","31031","109060","252909","391476","400452","262548","104390","23023","2457","104","1"],["28","677","7225","44804","178871","481327","888573","1124040","957372","529958","179894","34216","3185","119","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","10","33","105","324","977","2895","8462","24465","70101","199368","563425","1583643","4430290"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-48","63","-12"],["80","-66","8"],["-44","16","0"],["8","0","0"]]},"terms":[{"coeff":{"coeffs":[["-24","-42","175","-44"],["52","136","-182","24"],["-36","-108","48","0"],["8","24","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-6","-11","48","-16"],["0","10","32","-32","0"],["0","-4","-16","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172489,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-faf2521b7eb840ec"]},{"id":172490,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-faf2521b7eb840ec"]},{"id":172491,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-faf2521b7eb840ec"]}],"region_exact":{},"incoming_edges":[{"id":"e-faf2521b7eb840ec","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-26e4387737b6859f","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":"0312f35afa902a6a27f938979418e455ddb325127053da0f4f5d4f2c1a7f044c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d8cfb1e39c5d56b0","role":"sequence","name":"reversal of Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-180549e5010ed723","kind":"fingerprint_related","node_a_id":"n-26e4387737b6859f","node_b_id":"n-f783cab6fae5f997","fingerprint":"f6-285a8f351a67f82b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-285a8f351a67f82b"},"other_id":"n-f783cab6fae5f997","other":{"id":"n-f783cab6fae5f997","name":"one_plus_derivative of basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","first_index":0,"fingerprint":"f6-285a8f351a67f82b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["8","4"],["64","80","16"]],"card_rows":[["1"],["8","4"],["64","80","16"],["384","1088","576","64"]],"row_sums_preview":["1","12","160","2112","26880","331776","4001792","47431680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c30cb06ad75c5e21","steps":["basis A013611","D+1"],"edges":["e-36be6b9e5c42de00","e-2d6654014919feb8"],"text":"A103327 \u2192 basis A013611 \u2192 D+1"},"headline":"A103327 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-62ba22204c7f9797","kind":"fingerprint_related","node_a_id":"n-9887e767efe7cad8","node_b_id":"n-26e4387737b6859f","fingerprint":"f6-285a8f351a67f82b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-285a8f351a67f82b"},"other_id":"n-9887e767efe7cad8","other":{"id":"n-9887e767efe7cad8","name":"one_plus_derivative of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","first_index":0,"fingerprint":"f6-285a8f351a67f82b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["4","5","1"]],"card_rows":[["1"],["2","1"],["4","5","1"],["6","17","9","1"]],"row_sums_preview":["1","3","10","33","105","324","977","2895"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6758fe7729845be8","steps":["D+1"],"edges":["e-eb6017c7ac41e928"],"text":"A054142 \u2192 D+1"},"headline":"A054142 \u2192 D+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2:  2,   1;\n   3:  4,   5,    1;\n   4:  6,  17,    9,    1;\n   5:  8,  37,   45,   14,    1;\n   6: 10,  65,  133,   95,   20,    1;\n   7: 12, 101,  297,  364,  175,   27,    1;\n   8: 14, 145,  561, 1005,  840,  294,   35,   1;\n   9: 16, 197,  949, 2266, 2805, 1722,  462,  44,  1;\n  10: 18, 257, 1485, 4459, 7436, 6831, 3234, 690, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-44*n^2+80*n-48+(16*n^2-66*n+63)*x+(8*n-12)*x^2)*P(n,x) = (8*n^3-36*n^2+52*n-24+(24*n^3-108*n^2+136*n-42)*x+(48*n^2-182*n+175)*x^2+(24*n-44)*x^3)*P(n-1,x) - ((4*n^2-10*n+6)*x+(16*n^2-32*n+11)*x^2+(32*n-48)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 36*n^2 + 52*n - 24 + (24*n^3 - 108*n^2 + 136*n - 42)*x + (48*n^2 - 182*n + 175)*x^2 + (24*n - 44)*x^3)*P[n - 1, x] - ((4*n^2 - 10*n + 6)*x + (16*n^2 - 32*n + 11)*x^2 + (32*n - 48)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 44*n^2 + 80*n - 48 + (16*n^2 - 66*n + 63)*x + (8*n - 12)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^3 - 36*n^2 + 52*n - 24 + (24*n^3 - 108*n^2 + 136*n - 42)*x + (48*n^2 - 182*n + 175)*x^2 + (24*n - 44)*x^3)*v[i - 1] - ((4*n^2 - 10*n + 6)*x + (16*n^2 - 32*n + 11)*x^2 + (32*n - 48)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 44*n^2 + 80*n - 48 + (16*n^2 - 66*n + 63)*x + (8*n - 12)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-0eba401b117f16a9","steps":["reverse","D+1"],"edges":["e-6116c33129f68e6c","e-faf2521b7eb840ec"],"text":"A085478 \u2192 reverse \u2192 D+1"},"headline":"A085478 \u2192 reverse \u2192 D+1","chain":null}
