{"id":"n-3347371a4f01b6c3","revision":0,"first_index":0,"fingerprint":"f6-d60dbc403e0c506a","name":"derivative of gamma_polynomial of Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["3"],["15"],["45","18"],["105","126"],["210","504","84"],["378","1512","756"],["630","3780","3780","360"],["990","8316","13860","3960"],["1485","16632","41580","23760","1485"],["2145","30888","108108","102960","19305"],["3003","54054","252252","360360","135135","6006"],["4095","90090","540540","1081080","675675","90090"],["5460","144144","1081080","2882880","2702700","720720","24024"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","3","15","63","231","798","2646","8550","27126","84942","263406","810810","2481570","7561008"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["3"]],"recurrence":{"denominator":{"coeffs":[["-12"],["4"],["1"]]},"terms":[{"coeff":{"coeffs":[["12","-24"],["8","-4"],["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-16"],["0","2","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173783,"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-507468489ee05a91"]},{"id":173784,"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-507468489ee05a91"]},{"id":173785,"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-507468489ee05a91"]}],"region_exact":{},"incoming_edges":[{"id":"e-507468489ee05a91","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-3347371a4f01b6c3","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":"24795913b39e90a80455644ac6f9b78c274886e1bc3b65e528fc061978c7848a","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-92efd0b8c392ac45","role":"sequence","name":"gamma_polynomial of Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.","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:   0;\n  2:   3;\n  3:  15;\n  4:  45,   18;\n  5: 105,  126;\n  6: 210,  504,    84;\n  7: 378, 1512,   756;\n  8: 630, 3780,  3780,  360;\n  9: 990, 8316, 13860, 3960;\n","recurrence_sentence":"The row polynomials satisfy (n^2+4*n-12)*P(n,x) = (n^2+8*n+12+(4*n^2-4*n-24)*x)*P(n-1,x) + ((2*n+4)*x-(8*n+16)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 8*n + 12 + (4*n^2 - 4*n - 24)*x)*P[n - 1, x] + ((2*n + 4)*x - (8*n + 16)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n - 12)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 3; for(n = 3, N, my(i = n + 1); v[i] = ((n^2 + 8*n + 12 + (4*n^2 - 4*n - 24)*x)*v[i - 1] + ((2*n + 4)*x - (8*n + 16)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n - 12)); 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-bb74f03367de3ec8","steps":["\u03b3","D"],"edges":["e-fbbe2b618491014f","e-507468489ee05a91"],"text":"A145597 \u2192 \u03b3 \u2192 D"},"headline":"A145597 \u2192 \u03b3 \u2192 D","chain":null}