{"id":"n-6057763109e64c30","revision":0,"first_index":0,"fingerprint":"f6-a54a57f9f75c33a2","name":"one_plus_derivative of shift_x_plus_one of Riordan array (1+2x,x(1+x)).","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"],["4","1"],["9","7","1"],["20","27","10","1"],["40","81","53","13","1"],["78","210","204","87","16","1"],["147","498","649","409","129","19","1"],["272","1111","1826","1545","716","179","22","1"],["495","2371","4717","5043","3134","1145","237","25","1"],["890","4893","11444","14839","11658","5700","1716","303","28","1"],["1584","9835","26464","40417","38551","23820","9575","2449","377","31","1"],["2796","19354","58924","103704","116644","87535","44430","15139","3364","459","34","1"],["4901","37428","127242","253750","329286","291595","179745","77253","22820","4481","549","37","1"],["8540","71333","267912","597354","879314","899060","654160","341327","127030","33094","5820","647","40","1"],["14805","134279","552257","1361862","2243579","2604200","2188845","1348145","608795","199590","46485","7401","753","43","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","17","58","188","596","1852","5672","17168","51472","153104","452384","1329088","3885632","11311040"],"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","0","0","0"],["-5","-4","4","1"],["1","5","1","0"],["1","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["-15","-18","22","10","1"],["-4","26","1","-1","0"],["9","-7","-2","0","0"],["-2","0","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","15","48","44","12","1"],["5","31","36","11","1","0"],["5","6","1","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173579,"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-cf6769350f25b6c9"]},{"id":173580,"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-cf6769350f25b6c9"]},{"id":173581,"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-cf6769350f25b6c9"]}],"region_exact":{},"incoming_edges":[{"id":"e-cf6769350f25b6c9","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-6057763109e64c30","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":"bdf55e51f37e224040612f3b09722e51f01c1347b30afbacbfa67c4eaa9a9900","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-fb2cd7d8c683bee9","role":"sequence","name":"shift_x_plus_one of Riordan array (1+2x,x(1+x)).","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:   4,    1;\n  2:   9,    7,     1;\n  3:  20,   27,    10,     1;\n  4:  40,   81,    53,    13,     1;\n  5:  78,  210,   204,    87,    16,    1;\n  6: 147,  498,   649,   409,   129,   19,    1;\n  7: 272, 1111,  1826,  1545,   716,  179,   22,   1;\n  8: 495, 2371,  4717,  5043,  3134, 1145,  237,  25,  1;\n  9: 890, 4893, 11444, 14839, 11658, 5700, 1716, 303, 28, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^3+n^2-5*n+(5*n^2-4*n)*x+(n^2+4*n)*x^2+n*x^3)*P(n,x) = -(2*n^3-9*n^2+4*n+15+(7*n^2-26*n+18)*x+(2*n^2-n-22)*x^2+(n-10)*x^3-x^4)*P(n-1,x) + (5*n^2+5*n+(6*n^2+31*n+15)*x+(n^2+36*n+48)*x^2+(11*n+44)*x^3+(n+12)*x^4+x^5)*d/dx 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[(-(2*n^3 - 9*n^2 + 4*n + 15 + (7*n^2 - 26*n + 18)*x + (2*n^2 - n - 22)*x^2 + (n - 10)*x^3 - x^4)*P[n - 1, x] + (5*n^2 + 5*n + (6*n^2 + 31*n + 15)*x + (n^2 + 36*n + 48)*x^2 + (11*n + 44)*x^3 + (n + 12)*x^4 + x^5)*D[P[n - 1, x], x])/(n^3 + n^2 - 5*n + (5*n^2 - 4*n)*x + (n^2 + 4*n)*x^2 + n*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] = (-(2*n^3 - 9*n^2 + 4*n + 15 + (7*n^2 - 26*n + 18)*x + (2*n^2 - n - 22)*x^2 + (n - 10)*x^3 - x^4)*v[i - 1] + (5*n^2 + 5*n + (6*n^2 + 31*n + 15)*x + (n^2 + 36*n + 48)*x^2 + (11*n + 44)*x^3 + (n + 12)*x^4 + x^5)*deriv(v[i - 1], x))/(n^3 + n^2 - 5*n + (5*n^2 - 4*n)*x + (n^2 + 4*n)*x^2 + n*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-c27b11147fce8bfa","steps":["basis A007318","D+1"],"edges":["e-3cf676fe863061a7","e-cf6769350f25b6c9"],"text":"A113214 \u2192 basis A007318 \u2192 D+1"},"headline":"A113214 \u2192 basis A007318 \u2192 D+1","chain":null}