{"id":"n-6b61a305cbd15475","revision":1,"first_index":0,"fingerprint":"f6-64cbfb83c47eb75f","name":"one_plus_derivative of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["5","3"],["12","21","6"],["22","66","54","10"],["35","150","210","110","15"],["51","285","570","510","195","21"],["70","483","1260","1610","1050","315","28"],["92","756","2436","4060","3780","1932","476","36"],["117","1116","4284","8820","10710","7812","3276","684","45"],["145","1575","7020","17220","25830","24570","14700","5220","945","55"],["176","2145","10890","31020","55440","65142","50820","25740","7920","1265","66"],["210","2838","16170","52470","108900","152460","146916","97020","42570","11550","1650","78"],["247","3666","23166","84370","199485","324324","372372","303732","173745","67210","16302","2106","91"],["287","4641","32214","130130","345345","639639","852852","828828","585585","295295","102102","22386","2639","105"],["330","5775","43680","193830","570570","1186185","1801800","2033460","1711710","1066065","480480","150150","30030","3255","120"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","39","152","520","1632","4816","13568","36864","97280","250624","632832","1570816","3842048","9277440"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","3"],["12","21","6"],["22","66","54","10"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-4","-6","-4","-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["4","12","12","4"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-6","-12","-6"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163907,"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-110d52061e80882d"]},{"id":163908,"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-110d52061e80882d"]},{"id":163909,"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-110d52061e80882d"]}],"region_exact":{},"incoming_edges":[{"id":"e-110d52061e80882d","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-6b61a305cbd15475","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":"cd1a4c9f73b08b5976af7aac0f3a781ff4270a8c2e56600f0c50678134369675","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-787407680a56b31c","role":"sequence","name":"Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","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:   1;\n  1:   5,    3;\n  2:  12,   21,    6;\n  3:  22,   66,   54,    10;\n  4:  35,  150,  210,   110,    15;\n  5:  51,  285,  570,   510,   195,    21;\n  6:  70,  483, 1260,  1610,  1050,   315,    28;\n  7:  92,  756, 2436,  4060,  3780,  1932,   476,   36;\n  8: 117, 1116, 4284,  8820, 10710,  7812,  3276,  684,  45;\n  9: 145, 1575, 7020, 17220, 25830, 24570, 14700, 5220, 945, 55;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+4*x+6*x^2+4*x^3+x^4)*P(n-4,x) + (4+12*x+12*x^2+4*x^3)*P(n-3,x) - (6+12*x+6*x^2)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 5 + 3*x, P(2,x) = 12 + 21*x + 6*x^2, P(3,x) = 22 + 66*x + 54*x^2 + 10*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 3*x;\nP[2, x_] := 12 + 21*x + 6*x^2;\nP[3, x_] := 22 + 66*x + 54*x^2 + 10*x^3;\nP[n_, x_] := P[n, x] = Expand[-(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*P[n - 4, x] + (4 + 12*x + 12*x^2 + 4*x^3)*P[n - 3, x] - (6 + 12*x + 6*x^2)*P[n - 2, x] + (4 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 3*x; v[3] = 12 + 21*x + 6*x^2; v[4] = 22 + 66*x + 54*x^2 + 10*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*v[i - 4] + (4 + 12*x + 12*x^2 + 4*x^3)*v[i - 3] - (6 + 12*x + 6*x^2)*v[i - 2] + (4 + 4*x)*v[i - 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-c8c19eba9d68881f","steps":["(1/2+(1/2)n)\u03b8+1+n","D+1"],"edges":["e-c94c2672fb190cd9","e-110d52061e80882d"],"text":"A007318 \u2192 (1/2+(1/2)n)\u03b8+1+n \u2192 D+1"},"headline":"A007318 \u2192 (1/2+(1/2)n)\u03b8+1+n \u2192 D+1","chain":null}