{"id":"n-1b336857eed4ab2c","revision":1,"first_index":1,"fingerprint":"f6-d90d5bfb2679d7c4","name":"derivative of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["4","2"],["10","12","3"],["20","42","24","4"],["35","112","108","40","5"],["56","252","360","220","60","6"],["84","504","990","880","390","84","7"],["120","924","2376","2860","1820","630","112","8"],["165","1584","5148","8008","6825","3360","952","144","9"],["220","2574","10296","20020","21840","14280","5712","1368","180","10"],["286","4004","19305","45760","61880","51408","27132","9120","1890","220","11"],["364","6006","34320","97240","159120","162792","108528","47880","13860","2530","264","12"],["455","8736","58344","194480","377910","465120","379848","210672","79695","20240","3300","312","13"],["560","12376","95472","369512","839800","1220940","1193808","807576","382536","126500","28600","4212","364","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","6","25","90","300","954","2939","8850","26195","76500","221016","632916","1799125","5082270"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-4"],["2"]]},"terms":[{"coeff":{"coeffs":[["2","1"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163581,"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-1dbefba2139b076a"]},{"id":163582,"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-1dbefba2139b076a"]},{"id":163583,"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-1dbefba2139b076a"]},{"id":163584,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-1dbefba2139b076a"]}],"region_exact":{},"incoming_edges":[{"id":"e-1dbefba2139b076a","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-1b336857eed4ab2c","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":"b518abfb8233aa412187f133ade6f7ee6b09361668bde3e4bdadf8643c2c537a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-976cdb6e4a693ec8","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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-e56b0c54d4aaca32","kind":"oeis_match","node_a_id":"n-1b336857eed4ab2c","node_b_id":null,"fingerprint":null,"oeis_id":"A001871","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A001871","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001871","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-1b336857eed4ab2c","oeis_id":"A001871","target":"row_sums","oeis_offset":0,"first_row":2,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":26,"oeis_terms_available":26,"term_count":200,"raw_size":9709}],"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:   0;\n   2:   1;\n   3:   4,    2;\n   4:  10,   12,    3;\n   5:  20,   42,   24,    4;\n   6:  35,  112,  108,   40,    5;\n   7:  56,  252,  360,  220,   60,    6;\n   8:  84,  504,  990,  880,  390,   84,   7;\n   9: 120,  924, 2376, 2860, 1820,  630, 112,   8;\n  10: 165, 1584, 5148, 8008, 6825, 3360, 952, 144, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n-4)*P(n,x) = (2*n+2+(n+1)*x)*P(n-1,x) + (4*x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (n + 1)*x)*P[n - 1, x] + (4*x + x^2)*D[P[n - 1, x], x])/(2*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((2*n + 2 + (n + 1)*x)*v[i - 1] + (4*x + x^2)*deriv(v[i - 1], x))/(2*n - 4)); 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-976cdb6e4a693ec8","steps":["D"],"edges":["e-1dbefba2139b076a"],"text":"A078812 \u2192 D"},"headline":"A078812 \u2192 D","chain":null}