{"id":"n-b0113a98618fdee8","revision":0,"first_index":0,"fingerprint":"f6-d997120effc7e338","name":"one_plus_derivative of Triangle read by rows. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 - n], [2, 3], -x) for n >= 1.","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"],["1","1"],["1","3","1"],["1","9","7","1"],["1","21","40","14","1"],["1","41","170","130","25","1"],["1","71","560","875","350","41","1"],["1","113","1526","4410","3430","826","63","1"],["1","169","3612","17640","24696","11172","1764","92","1"],["1","241","7680","58968","137592","109368","31752","3480","129","1"],["1","331","15015","171270","623700","814968","407484","81180","6435","175","1"],["1","441","27445","444675","2395800","4848228","3933468","1328580","190575","11275","231","1"],["1","573","47476","1053910","8045895","23981958","29899584","16195608","3893175","416845","18876","298","1"],["1","729","78442","2316314","24191167","101897367","186249492","153858276","58709079","10449439","858858","30394","377","1"],["1","911","124670","4778137","66314248","381476095","982533552","1184303835","684264438","191668477","26052026","1681043","47320","469","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","5","18","77","368","1899","10370","59147","349212","2120560","13180720","83554200","538639936","3523245223"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","-18","-12","6"],["-6","-33","8","5"],["-5","0","13","1"],["5","12","3","0"],["5","3","0","0"],["1","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","18","21","0","-3"],["-6","6","-20","-12","5"],["-5","-47","-55","6","10"],["5","-31","0","30","0"],["5","8","30","0","0"],["1","10","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-36","-69","-27","9","3"],["0","-15","37","79","12","-15"],["0","27","83","17","-45","0"],["0","21","0","-45","0","0"],["0","3","-15","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-12","-24","-6","12","6"],["0","0","-4","8","30","18","0"],["0","0","10","28","18","0","0"],["0","0","6","6","0","0","0"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174858,"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-edda87f4c69a7295"]},{"id":174859,"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-edda87f4c69a7295"]},{"id":174860,"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-edda87f4c69a7295"]}],"region_exact":{},"incoming_edges":[{"id":"e-edda87f4c69a7295","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-b0113a98618fdee8","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":"6e4151dc39e822b32c2eec4c45e8f45839bbc43aceefed561faafd650093821d","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-afb2f273dc5c03d1","role":"sequence","name":"Triangle read by rows. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 - n], [2, 3], -x) for n >= 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: 1;\n  1: 1,   1;\n  2: 1,   3,    1;\n  3: 1,   9,    7,     1;\n  4: 1,  21,   40,    14,      1;\n  5: 1,  41,  170,   130,     25,      1;\n  6: 1,  71,  560,   875,    350,     41,     1;\n  7: 1, 113, 1526,  4410,   3430,    826,    63,    1;\n  8: 1, 169, 3612, 17640,  24696,  11172,  1764,   92,   1;\n  9: 1, 241, 7680, 58968, 137592, 109368, 31752, 3480, 129, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^5+5*n^4+5*n^3-5*n^2-6*n+(3*n^4+12*n^3-33*n-18)*x+(3*n^3+13*n^2+8*n-12)*x^2+(n^2+5*n+6)*x^3)*P(n,x) = (n^5+5*n^4+5*n^3-5*n^2-6*n+(10*n^5+8*n^4-31*n^3-47*n^2+6*n+18)*x+(30*n^4-55*n^2-20*n+21)*x^2+(30*n^3+6*n^2-12*n)*x^3+(10*n^2+5*n-3)*x^4)*P(n-1,x) + ((3*n^4+21*n^3+27*n^2-15*n-36)*x-(15*n^4-83*n^2-37*n+69)*x^2-(45*n^3-17*n^2-79*n+27)*x^3-(45*n^2-12*n-9)*x^4-(15*n-3)*x^5)*d/dx P(n-1,x) + ((6*n^3+10*n^2-4*n-12)*x^2+(6*n^3+28*n^2+8*n-24)*x^3+(18*n^2+30*n-6)*x^4+(18*n+12)*x^5+6*x^6)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n + (10*n^5 + 8*n^4 - 31*n^3 - 47*n^2 + 6*n + 18)*x + (30*n^4 - 55*n^2 - 20*n + 21)*x^2 + (30*n^3 + 6*n^2 - 12*n)*x^3 + (10*n^2 + 5*n - 3)*x^4)*P[n - 1, x] + ((3*n^4 + 21*n^3 + 27*n^2 - 15*n - 36)*x - (15*n^4 - 83*n^2 - 37*n + 69)*x^2 - (45*n^3 - 17*n^2 - 79*n + 27)*x^3 - (45*n^2 - 12*n - 9)*x^4 - (15*n - 3)*x^5)*D[P[n - 1, x], x] + ((6*n^3 + 10*n^2 - 4*n - 12)*x^2 + (6*n^3 + 28*n^2 + 8*n - 24)*x^3 + (18*n^2 + 30*n - 6)*x^4 + (18*n + 12)*x^5 + 6*x^6)*D[P[n - 1, x], {x, 2}])/(n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n + (3*n^4 + 12*n^3 - 33*n - 18)*x + (3*n^3 + 13*n^2 + 8*n - 12)*x^2 + (n^2 + 5*n + 6)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n + (10*n^5 + 8*n^4 - 31*n^3 - 47*n^2 + 6*n + 18)*x + (30*n^4 - 55*n^2 - 20*n + 21)*x^2 + (30*n^3 + 6*n^2 - 12*n)*x^3 + (10*n^2 + 5*n - 3)*x^4)*v[i - 1] + ((3*n^4 + 21*n^3 + 27*n^2 - 15*n - 36)*x - (15*n^4 - 83*n^2 - 37*n + 69)*x^2 - (45*n^3 - 17*n^2 - 79*n + 27)*x^3 - (45*n^2 - 12*n - 9)*x^4 - (15*n - 3)*x^5)*deriv(v[i - 1], x) + ((6*n^3 + 10*n^2 - 4*n - 12)*x^2 + (6*n^3 + 28*n^2 + 8*n - 24)*x^3 + (18*n^2 + 30*n - 6)*x^4 + (18*n + 12)*x^5 + 6*x^6)*deriv(deriv(v[i - 1], x), x))/(n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n + (3*n^4 + 12*n^3 - 33*n - 18)*x + (3*n^3 + 13*n^2 + 8*n - 12)*x^2 + (n^2 + 5*n + 6)*x^3)); 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-afb2f273dc5c03d1","steps":["D+1"],"edges":["e-edda87f4c69a7295"],"text":"A359363 \u2192 D+1"},"headline":"A359363 \u2192 D+1","chain":null}