{"id":"n-8d8a4edbb9b84147","revision":0,"first_index":0,"fingerprint":"f6-983bc0c4ad0f2ee6","name":"derivative of gamma_polynomial of Pascal-(1,7,1) array.","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":[[],[],["7"],["14"],["21","98"],["28","294"],["35","588","1029"],["42","980","4116"],["49","1470","10290","9604"],["56","2058","20580","48020"],["63","2744","36015","144060","84035"],["70","3528","57624","336140","504210"],["77","4410","86436","672280","1764735","705894"],["84","5390","123480","1210104","4705960","4941258"],["91","6468","169785","2016840","10588410","19765032","5764801"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","7","14","119","322","1652","5138","21413","70714","266917","901572","3233832","10986276","38311427"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["7"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-42"],["1","14"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-28"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174318,"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-d36710f563c4e6fa"]},{"id":174319,"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-d36710f563c4e6fa"]},{"id":174320,"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-d36710f563c4e6fa"]}],"region_exact":{},"incoming_edges":[{"id":"e-d36710f563c4e6fa","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-8d8a4edbb9b84147","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":"3c71c529e168bdf8f980404c0f8a2c2ba72202efbe4febc9a2ecd8e33b7aed0c","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-695d6cb541611b1e","role":"sequence","name":"gamma_polynomial of Pascal-(1,7,1) array.","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:  7;\n  3: 14;\n  4: 21,   98;\n  5: 28,  294;\n  6: 35,  588,  1029;\n  7: 42,  980,  4116;\n  8: 49, 1470, 10290,  9604;\n  9: 56, 2058, 20580, 48020;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (n-1+(14*n-42)*x)*P(n-1,x) - (x+28*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 7.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 7;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (14*n - 42)*x)*P[n - 1, x] - (x + 28*x^2)*D[P[n - 1, x], x])/(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] = 0; v[3] = 7; for(n = 3, N, my(i = n + 1); v[i] = ((n - 1 + (14*n - 42)*x)*v[i - 1] - (x + 28*x^2)*deriv(v[i - 1], x))/(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-1e2f1b7f566b371c","steps":["\u03b3","D"],"edges":["e-c807fb938e5557d1","e-d36710f563c4e6fa"],"text":"A081582 \u2192 \u03b3 \u2192 D"},"headline":"A081582 \u2192 \u03b3 \u2192 D","chain":null}