{"id":"n-a702c544a0c5fe42","revision":0,"first_index":0,"fingerprint":"f6-2d7d778b892c56a7","name":"one_plus_derivative of shift_x_plus_one of Triangle read by rows: coefficients d(n,k) of Andr\u00e9 polynomials D(x,n) = Sum_{k>0} d(n,k)*x^k.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["3","1"],["9","4"],["35","27","4"],["155","162","34"],["791","1083","384","34"],["4529","7656","3744","496"],["28839","58391","37560","8256","496"],["201939","477086","381810","117216","11056"],["1542739","4173519","4021100","1630394","252144","11056"],["12767689","38953308","44000600","22330568","4867128","349504"],["113794603","386812467","502255020","308239960","89064720","10407336","349504"],["1086657403","4074402282","5986276290","4324583200","1571760480","260573616","14873104"],["11068604847","45396768979","74524480392","62095575570","27397453840","6041929344","558624288","14873104"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","4","13","66","351","2292","16425","133542","1189107","11630952","123268797","1410923610","17319126375","227098310364"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["1","4"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["-2","-10","-12","8","16"],["5","21","26","4","-8"],["1","-2","-9","-6","0"],["-1","-2","-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2","14","36","40","16"],["2","13","31","32","12","0"],["1","4","5","2","0","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3","-15","-12","8"],["0","7","10","0"],["2","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-7","-14","-8"],["-1","-3","-2","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175854,"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-66e8d6ad62da43a0"]},{"id":175855,"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-66e8d6ad62da43a0"]},{"id":175856,"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-66e8d6ad62da43a0"]}],"region_exact":{},"incoming_edges":[{"id":"e-66e8d6ad62da43a0","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-a702c544a0c5fe42","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":"00295adbfeec72475ed1e2f966b6e1d93c9a66d55068617b15617e5d345697d2","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-180398bbec0732d0","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: coefficients d(n,k) of Andr\u00e9 polynomials D(x,n) = Sum_{k>0} d(n,k)*x^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:      1;\n  2:      3,      1;\n  3:      9,      4;\n  4:     35,     27,      4;\n  5:    155,    162,     34;\n  6:    791,   1083,    384,     34;\n  7:   4529,   7656,   3744,    496;\n  8:  28839,  58391,  37560,   8256,   496;\n  9: 201939, 477086, 381810, 117216, 11056;\n","recurrence_sentence":"The row polynomials satisfy (n+1+4*x)*P(n,x) = -(n^3-n^2-5*n+2+(2*n^3+2*n^2-21*n+10)*x+(n^3+9*n^2-26*n+12)*x^2+(6*n^2-4*n-8)*x^3+(8*n-16)*x^4)*P(n-2,x) + (n^2+2*n+(4*n^2+13*n+2)*x+(5*n^2+31*n+14)*x^2+(2*n^2+32*n+36)*x^3+(12*n+40)*x^4+16*x^5)*d/dx P(n-2,x) + (2*n^2-3+(2*n^2+7*n-15)*x+(10*n-12)*x^2+8*x^3)*P(n-1,x) - (n+1+(3*n+7)*x+(2*n+14)*x^2+8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2 - 5*n + 2 + (2*n^3 + 2*n^2 - 21*n + 10)*x + (n^3 + 9*n^2 - 26*n + 12)*x^2 + (6*n^2 - 4*n - 8)*x^3 + (8*n - 16)*x^4)*P[n - 2, x] + (n^2 + 2*n + (4*n^2 + 13*n + 2)*x + (5*n^2 + 31*n + 14)*x^2 + (2*n^2 + 32*n + 36)*x^3 + (12*n + 40)*x^4 + 16*x^5)*D[P[n - 2, x], x] + (2*n^2 - 3 + (2*n^2 + 7*n - 15)*x + (10*n - 12)*x^2 + 8*x^3)*P[n - 1, x] - (n + 1 + (3*n + 7)*x + (2*n + 14)*x^2 + 8*x^3)*D[P[n - 1, x], x])/(n + 1 + 4*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - n^2 - 5*n + 2 + (2*n^3 + 2*n^2 - 21*n + 10)*x + (n^3 + 9*n^2 - 26*n + 12)*x^2 + (6*n^2 - 4*n - 8)*x^3 + (8*n - 16)*x^4)*v[i - 2] + (n^2 + 2*n + (4*n^2 + 13*n + 2)*x + (5*n^2 + 31*n + 14)*x^2 + (2*n^2 + 32*n + 36)*x^3 + (12*n + 40)*x^4 + 16*x^5)*deriv(v[i - 2], x) + (2*n^2 - 3 + (2*n^2 + 7*n - 15)*x + (10*n - 12)*x^2 + 8*x^3)*v[i - 1] - (n + 1 + (3*n + 7)*x + (2*n + 14)*x^2 + 8*x^3)*deriv(v[i - 1], x))/(n + 1 + 4*x)); 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-ff270bcb802a85af","steps":["\u03b3 (scale 1/2) (shift +1)","x \u21a6 x+1","D+1"],"edges":["e-977fc5754e9861fd","e-6d784857f6230dda","e-66e8d6ad62da43a0"],"text":"A008292 \u2192 \u03b3 (scale 1/2) (shift +1) \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A008292 \u2192 \u03b3 (scale 1/2) (shift +1) \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}