{"id":"n-e9fa4da31855e6d0","revision":0,"first_index":0,"fingerprint":"f6-cc9f40497aa70b3e","name":"shift_x_plus_one of derivative of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1","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":false,"rows":[[],["2"],["14","6"],["72","64","12"],["330","450","180","20"],["1430","2640","1650","400","30"],["6006","14014","12012","4620","770","42"],["24752","69888","76440","40768","10920","1344","56"],["100776","334152","445536","305760","114660","22932","2184","72"],["406980","1550400","2441880","2056320","999600","282240","44100","3360","90"],["1634380","7034940","12790800","12790800","7674480","2827440","628320","79200","4950","110"],["6537520","31380096","64721448","75039360","53721360","24558336","7162848","1292544","134640","7040","132"],["26075790","138105110","318704100","420689412","350574510","192053862","69837768","16628040","2494206","218790","9724","156"],["103791870","601277040","1535403870","2274672400","2165313150","1385800416","606287682","180756576","35945910","4564560","342342","13104","182"],["412506150","2594796750","7265430900","11942030100","12795032250","9383023650","4811807000","1732250520","433062630","73223150","7987980","518700","17290","210"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","20","148","980","6150","37464","224168","1326072","7784970","45465420","264555324","1535391468","8894169102","51451687280"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-4"],["3"],["1"]]},"terms":[{"coeff":{"coeffs":[["6","-2"],["10","4"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-3"],["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176628,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-3979c4c1afc18548"]},{"id":176629,"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-0df0e03db0025767","e-3979c4c1afc18548"]},{"id":176630,"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-0df0e03db0025767","e-3979c4c1afc18548"]},{"id":176631,"position":3,"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-0df0e03db0025767"]}],"region_exact":{},"incoming_edges":[{"id":"e-0df0e03db0025767","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-e9fa4da31855e6d0","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":"2463c3116cff5075f2a926e218ca6aed2f6410c275e76ecf13de54d77d07b5fb","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-3c983d83f25bade9","role":"sequence","name":"shift_x_plus_one of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1","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]"}]},{"id":"e-3979c4c1afc18548","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-e9fa4da31855e6d0","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"bc5b9f5d0b936cc6b1dcb72d195d770093e777fc5c9ab052a810fda3450401e0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-7bf0ce499426260c","role":"sequence","name":"derivative of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1","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:      2;\n  2:     14,       6;\n  3:     72,      64,      12;\n  4:    330,     450,     180,      20;\n  5:   1430,    2640,    1650,     400,     30;\n  6:   6006,   14014,   12012,    4620,    770,     42;\n  7:  24752,   69888,   76440,   40768,  10920,   1344,    56;\n  8: 100776,  334152,  445536,  305760, 114660,  22932,  2184,   72;\n  9: 406980, 1550400, 2441880, 2056320, 999600, 282240, 44100, 3360, 90;\n","recurrence_sentence":"The row polynomials satisfy (n^2+3*n-4)*P(n,x) = (4*n^2+10*n+6+(3*n^2+4*n-2)*x)*P(n-1,x) - ((2*n+3)*x+(2*n+3)*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[((4*n^2 + 10*n + 6 + (3*n^2 + 4*n - 2)*x)*P[n - 1, x] - ((2*n + 3)*x + (2*n + 3)*x^2)*D[P[n - 1, x], x])/(n^2 + 3*n - 4)]];\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] = ((4*n^2 + 10*n + 6 + (3*n^2 + 4*n - 2)*x)*v[i - 1] - ((2*n + 3)*x + (2*n + 3)*x^2)*deriv(v[i - 1], x))/(n^2 + 3*n - 4)); 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-0e6d847331141ca6","steps":["x \u21a6 x+1","D"],"edges":["e-1d39f03bcbd1a8b9","e-0df0e03db0025767"],"text":"A145596 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A145596 \u2192 x \u21a6 x+1 \u2192 D","chain":null}