{"id":"n-f9ec10621819e3f8","revision":0,"first_index":0,"fingerprint":"f6-ffeaa4374b687cd3","name":"derivative of Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is the 1-tiered binomial coefficient.","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":true,"rows":[[],["3"],["8","14"],["17","62","45"],["34","192","282","124"],["67","516","1146","1012","315"],["132","1290","3840","5100","3180","762"],["261","3090","11535","20460","19155","9186","1785"],["518","7196","32298","71680","89530","64428","25046","4088"],["1031","16424","86100","229432","358330","343896","200564","65480","9207"],["2056","36918","221328","688296","1290240","1548036","1203888","589608","165816","20470"],["4105","81990","553185","1966440","4301010","6192900","6020490","3931560","1658565","409510","45045"],["8202","180312","1352010","5407380","13517460","22708224","26492004","21625560","12164130","4505160","991122","98292"],["16395","393324","3244494","14419020","40551885","77857560","105970788","103806648","72988245","36043260","11894190","2359164","212979"],["32780","852098","7668336","37488308","117148460","253037070","393609216","449835672","379546596","234286910","103085840","30670068","5537636","458738"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","22","124","632","3056","14304","65472","294784","1310464","5766656","25164800","109049856","469757952","2013257728"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["3"],["8","14"]],"recurrence":{"denominator":{"coeffs":[["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["-12","0","0"],["6","-14","2"],["0","9","-3"],["0","-1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4","-4","0","0"],["1","-3","-3","1"],["0","1","0","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176154,"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-88b1f14242293276"]},{"id":176155,"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-88b1f14242293276"]},{"id":176156,"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-88b1f14242293276"]}],"region_exact":{},"incoming_edges":[{"id":"e-88b1f14242293276","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-f9ec10621819e3f8","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":"ba084d3f9f8a7a94ed39db93faf6fc64a68d9a7f269ead58edc521c1c45c4ead","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-13e23eca5f474574","role":"sequence","name":"Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is the 1-tiered binomial coefficient.","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":[],"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:    3;\n  2:    8,    14;\n  3:   17,    62,    45;\n  4:   34,   192,   282,    124;\n  5:   67,   516,  1146,   1012,    315;\n  6:  132,  1290,  3840,   5100,   3180,    762;\n  7:  261,  3090, 11535,  20460,  19155,   9186,   1785;\n  8:  518,  7196, 32298,  71680,  89530,  64428,  25046,  4088;\n  9: 1031, 16424, 86100, 229432, 358330, 343896, 200564, 65480, 9207;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2)*P(n,x) = (6*n-12-(n^3-9*n^2+14*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-1,x) + (n-4+(n^2-3*n-4)*x-3*n*x^2-(n^2-n)*x^3)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 3, P(2,x) = 8 + 14*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[2, x_] := 8 + 14*x;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n - 12 - (n^3 - 9*n^2 + 14*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 1, x] + (n - 4 + (n^2 - 3*n - 4)*x - 3*n*x^2 - (n^2 - n)*x^3)*D[P[n - 1, x], x])/(n^2 - 3*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] = 3; v[3] = 8 + 14*x; for(n = 3, N, my(i = n + 1); v[i] = ((6*n - 12 - (n^3 - 9*n^2 + 14*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 1] + (n - 4 + (n^2 - 3*n - 4)*x - 3*n*x^2 - (n^2 - n)*x^3)*deriv(v[i - 1], x))/(n^2 - 3*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-13e23eca5f474574","steps":["D"],"edges":["e-88b1f14242293276"],"text":"A308737 \u2192 D"},"headline":"A308737 \u2192 D","chain":null}