{"id":"n-a6123995980aa3d8","revision":1,"first_index":0,"fingerprint":"f6-b9c4c96a61675307","name":"one_plus_derivative of Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).","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"],["4","2"],["11","14","3"],["22","54","30","4"],["37","140","150","52","5"],["56","290","480","320","80","6"],["79","522","1185","1220","585","114","7"],["106","854","2478","3500","2590","966","154","8"],["137","1304","4620","8344","8470","4872","1484","200","9"],["172","1890","7920","17472","22680","17892","8400","2160","252","10"],["211","2630","12735","33240","52710","53172","34230","13560","3015","310","11"],["254","3542","19470","58740","110220","135828","111804","60720","20790","4070","374","12"],["301","4644","28578","97900","212355","309672","311388","216216","101475","30580","5346","444","13"],["352","5954","40560","155584","383240","646074","768768","652080","391248","161590","43472","6864","520","14"],["407","7490","55965","237692","655655","1255254","1726725","1733160","1270269","670670","247247","60060","8645","602","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","28","110","384","1232","3712","10656","29440","78848","205824","525824","1318912","3256320","7929856"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","2"],["11","14","3"],["22","54","30","4"]],"recurrence":{"denominator":{"coeffs":[["0"],["-12"],["-14"],["12"],["7"],["-6"],["1"]]},"terms":[{"coeff":{"coeffs":[["36","120","144","72","12"],["6","24","36","24","6"],["-14","-48","-60","-32","-6"],["15","45","45","15","0"],["-5","-15","-15","-5","0"],["-3","-9","-9","-3","0"],["1","3","3","1","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["12","48","72","48","12"],["6","24","36","24","6"],["-6","-24","-36","-24","-6"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-24","-72","-72","-24"],["12","0","-36","-24"],["18","48","42","12"],["-60","-120","-60","0"],["9","18","9","0"],["12","24","12","0"],["-3","-6","-3","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-24","-72","-72","-24"],["-24","-72","-72","-24"],["12","36","36","12"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["-54","-36","18"],["-36","-42","-6"],["63","63","0"],["3","3","0"],["-15","-15","0"],["3","3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["18","36","18"],["-6","-12","-6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163898,"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-ac1cdf02143513b6"]},{"id":163899,"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-ac1cdf02143513b6"]},{"id":163900,"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-ac1cdf02143513b6"]}],"region_exact":{},"incoming_edges":[{"id":"e-ac1cdf02143513b6","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-a6123995980aa3d8","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":"ffa716afb5b68743146277a3c430b1e6c9de0b83a8e328d2fef917a8238dcb54","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-14d5814e0f152cf3","role":"sequence","name":"Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).","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:   4,    2;\n  2:  11,   14,    3;\n  3:  22,   54,   30,     4;\n  4:  37,  140,  150,    52,     5;\n  5:  56,  290,  480,   320,    80,     6;\n  6:  79,  522, 1185,  1220,   585,   114,    7;\n  7: 106,  854, 2478,  3500,  2590,   966,  154,    8;\n  8: 137, 1304, 4620,  8344,  8470,  4872, 1484,  200,   9;\n  9: 172, 1890, 7920, 17472, 22680, 17892, 8400, 2160, 252, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^6-6*n^5+7*n^4+12*n^3-14*n^2-12*n)*P(n,x) = (n^6-3*n^5-5*n^4+15*n^3-14*n^2+6*n+36+(3*n^6-9*n^5-15*n^4+45*n^3-48*n^2+24*n+120)*x+(3*n^6-9*n^5-15*n^4+45*n^3-60*n^2+36*n+144)*x^2+(n^6-3*n^5-5*n^4+15*n^3-32*n^2+24*n+72)*x^3-(6*n^2-6*n-12)*x^4)*P(n-3,x) - (6*n^2-6*n-12+(24*n^2-24*n-48)*x+(36*n^2-36*n-72)*x^2+(24*n^2-24*n-48)*x^3+(6*n^2-6*n-12)*x^4)*d/dx P(n-3,x) - (3*n^6-12*n^5-9*n^4+60*n^3-18*n^2-12*n+24+(6*n^6-24*n^5-18*n^4+120*n^3-48*n^2+72)*x+(3*n^6-12*n^5-9*n^4+60*n^3-42*n^2+36*n+72)*x^2-(12*n^2-24*n-24)*x^3)*P(n-2,x) + (12*n^2-24*n-24+(36*n^2-72*n-72)*x+(36*n^2-72*n-72)*x^2+(12*n^2-24*n-24)*x^3)*d/dx P(n-2,x) + (3*n^6-15*n^5+3*n^4+63*n^3-36*n^2-54*n+(3*n^6-15*n^5+3*n^4+63*n^3-42*n^2-36*n)*x-(6*n^2-18*n)*x^2)*P(n-1,x) - (6*n^2-18*n+(12*n^2-36*n)*x+(6*n^2-18*n)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4 + 2*x, P(2,x) = 11 + 14*x + 3*x^2, P(3,x) = 22 + 54*x + 30*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[2, x_] := 11 + 14*x + 3*x^2;\nP[3, x_] := 22 + 54*x + 30*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 14*n^2 + 6*n + 36 + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 48*n^2 + 24*n + 120)*x + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 60*n^2 + 36*n + 144)*x^2 + (n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 32*n^2 + 24*n + 72)*x^3 - (6*n^2 - 6*n - 12)*x^4)*P[n - 3, x] - (6*n^2 - 6*n - 12 + (24*n^2 - 24*n - 48)*x + (36*n^2 - 36*n - 72)*x^2 + (24*n^2 - 24*n - 48)*x^3 + (6*n^2 - 6*n - 12)*x^4)*D[P[n - 3, x], x] - (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 18*n^2 - 12*n + 24 + (6*n^6 - 24*n^5 - 18*n^4 + 120*n^3 - 48*n^2 + 72)*x + (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 42*n^2 + 36*n + 72)*x^2 - (12*n^2 - 24*n - 24)*x^3)*P[n - 2, x] + (12*n^2 - 24*n - 24 + (36*n^2 - 72*n - 72)*x + (36*n^2 - 72*n - 72)*x^2 + (12*n^2 - 24*n - 24)*x^3)*D[P[n - 2, x], x] + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 36*n^2 - 54*n + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 42*n^2 - 36*n)*x - (6*n^2 - 18*n)*x^2)*P[n - 1, x] - (6*n^2 - 18*n + (12*n^2 - 36*n)*x + (6*n^2 - 18*n)*x^2)*D[P[n - 1, x], x])/(n^6 - 6*n^5 + 7*n^4 + 12*n^3 - 14*n^2 - 12*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; v[3] = 11 + 14*x + 3*x^2; v[4] = 22 + 54*x + 30*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 14*n^2 + 6*n + 36 + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 48*n^2 + 24*n + 120)*x + (3*n^6 - 9*n^5 - 15*n^4 + 45*n^3 - 60*n^2 + 36*n + 144)*x^2 + (n^6 - 3*n^5 - 5*n^4 + 15*n^3 - 32*n^2 + 24*n + 72)*x^3 - (6*n^2 - 6*n - 12)*x^4)*v[i - 3] - (6*n^2 - 6*n - 12 + (24*n^2 - 24*n - 48)*x + (36*n^2 - 36*n - 72)*x^2 + (24*n^2 - 24*n - 48)*x^3 + (6*n^2 - 6*n - 12)*x^4)*deriv(v[i - 3], x) - (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 18*n^2 - 12*n + 24 + (6*n^6 - 24*n^5 - 18*n^4 + 120*n^3 - 48*n^2 + 72)*x + (3*n^6 - 12*n^5 - 9*n^4 + 60*n^3 - 42*n^2 + 36*n + 72)*x^2 - (12*n^2 - 24*n - 24)*x^3)*v[i - 2] + (12*n^2 - 24*n - 24 + (36*n^2 - 72*n - 72)*x + (36*n^2 - 72*n - 72)*x^2 + (12*n^2 - 24*n - 24)*x^3)*deriv(v[i - 2], x) + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 36*n^2 - 54*n + (3*n^6 - 15*n^5 + 3*n^4 + 63*n^3 - 42*n^2 - 36*n)*x - (6*n^2 - 18*n)*x^2)*v[i - 1] - (6*n^2 - 18*n + (12*n^2 - 36*n)*x + (6*n^2 - 18*n)*x^2)*deriv(v[i - 1], x))/(n^6 - 6*n^5 + 7*n^4 + 12*n^3 - 14*n^2 - 12*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-c8c19eba9d68881f","steps":["\u2212D\u00b2+(\u22121+n)D+1+n","D+1"],"edges":["e-168270346f6f7c93","e-ac1cdf02143513b6"],"text":"A007318 \u2192 \u2212D\u00b2+(\u22121+n)D+1+n \u2192 D+1"},"headline":"A007318 \u2192 \u2212D\u00b2+(\u22121+n)D+1+n \u2192 D+1","chain":null}