{"id":"n-7b7a9d67563ccd22","revision":1,"first_index":0,"fingerprint":"f6-c0201d95edde5fbe","name":"derivative of basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[[],["2"],["3","6"],["-12","24","12"],["-20","-40","90","20"],["60","-240","0","240","30"],["105","210","-1155","420","525","42"],["-280","1680","-840","-3360","1960","1008","56"],["-504","-1008","9828","-9576","-6300","6048","1764","72"],["1260","-10080","10080","30240","-44100","-5040","15120","2880","90"],["2310","4620","-69300","110880","36960","-138600","16170","33000","4455","110"],["-5544","55440","-83160","-221760","554400","-133056","-332640","95040","65340","6600","132"],["-10296","-20592","437580","-978120","-51480","1812096","-1009008","-617760","308880","120120","9438","156"],["24024","-288288","576576","1441440","-5345340","3171168","4036032","-3843840","-792792","800800","208208","13104","182"],["45045","90090","-2567565","7387380","-1576575","-18108090","18963945","4684680","-10945935","-150150","1816815","343980","17745","210"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","9","24","50","90","147","224","324","450","605","792","1014","1274","1575"],"nonn":false,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["0"],["-2"],["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["4","-2"],["8","-3"],["2","2"],["-2","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","3","-1"],["-6","9","-3"],["-4","6","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163980,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-d29f61265fea8d1e"]}],"region_exact":{},"incoming_edges":[{"id":"e-d29f61265fea8d1e","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-7b7a9d67563ccd22","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":"cba8196b251e4063d679e54ec2ff110dd48f3d0645a4e11636d8a17a5b76c67b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-3bc20048ee6f0787","role":"sequence","name":"basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-27eeeab6acbea944","kind":"oeis_match","node_a_id":"n-7b7a9d67563ccd22","node_b_id":null,"fingerprint":null,"oeis_id":"A006002","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A006002","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A006002","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-7b7a9d67563ccd22","oeis_id":"A006002","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":44,"oeis_terms_available":44,"term_count":200,"raw_size":2058}],"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:    2;\n  2:    3,      6;\n  3:  -12,     24,    12;\n  4:  -20,    -40,    90,    20;\n  5:   60,   -240,     0,   240,     30;\n  6:  105,    210, -1155,   420,    525,    42;\n  7: -280,   1680,  -840, -3360,   1960,  1008,    56;\n  8: -504,  -1008,  9828, -9576,  -6300,  6048,  1764,   72;\n  9: 1260, -10080, 10080, 30240, -44100, -5040, 15120, 2880, 90;\n","recurrence_sentence":"The row polynomials satisfy (n^3+n^2-2*n)*P(n,x) = -(2*n^3-2*n^2-8*n-4-(3*n^3+2*n^2-3*n-2)*x)*P(n-1,x) - (4*n^2+6*n+2-(6*n^2+9*n+3)*x+(2*n^2+3*n+1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 2*n^2 - 8*n - 4 - (3*n^3 + 2*n^2 - 3*n - 2)*x)*P[n - 1, x] - (4*n^2 + 6*n + 2 - (6*n^2 + 9*n + 3)*x + (2*n^2 + 3*n + 1)*x^2)*D[P[n - 1, x], x])/(n^3 + n^2 - 2*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^3 - 2*n^2 - 8*n - 4 - (3*n^3 + 2*n^2 - 3*n - 2)*x)*v[i - 1] - (4*n^2 + 6*n + 2 - (6*n^2 + 9*n + 3)*x + (2*n^2 + 3*n + 1)*x^2)*deriv(v[i - 1], x))/(n^3 + n^2 - 2*n)); 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-3bc20048ee6f0787","steps":["D"],"edges":["e-d29f61265fea8d1e"],"text":"basis-map coefficients of A132812 \u2192 D"},"headline":"basis-map coefficients of A132812 \u2192 D","chain":null}