{"id":"n-f3c39411cf79b61b","revision":0,"first_index":0,"fingerprint":"f6-e0453429b9960a60","name":"one_plus_derivative of derivative of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k=0 and 0<=k<=2n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["5","2"],["28","58","30","4"],["69","330","495","300","75","6"],["128","980","2772","3660","2460","852","140","8"],["205","2170","9210","20200","24915","17970","7630","1840","225","10"],["300","4062","23130","72240","135960","160770","121926","59520","18360","3390","330","12"],["413","6818","48825","199164","511980","869484","999229","786520","424872","155820","37653","5628","455","14"],["544","10600","91560","462280","1516200","3401328","5364800","6040280","4892040","2853200","1190112","348600","69160","8680","600","16"],["693","15570","157572","949680","3799530","10657836","21609000","32233824","35738415","29603790","18332424","8447040","2861586","696780","117180","12672","765","18"],["860","21890","254070","1781100","8423280","28445220","70870380","132870840","189742500","207874710","175355730","113939820","56832360","21581700","6145740","1279440","186660","17730","950","20"],["1045","29722","389235","3112780","17003745","67259610","199376562","452464320","796911885","1098818490","1192412925","1021092732","690130155","367218390","152914410","49304640","12099087","2198790","283195","23980","1155","22"],["1248","39228","572220","5142324","31878660","144689028","498181068","1330979760","2800541304","4686875820","6279171228","6761319228","5863232700","4095150444","2299237380","1033087968","368603928","103163940","22228404","3583140","413028","31548","1380","24"],["1469","50570","813150","8113560","56295525","288523950","1132801670","3488342000","8559489510","16917486300","27132701310","35486797320","37966237725","33277627110","23897817450","14039419680","6724073070","2610164700","813823010","201086600","38603565","5589870","583050","40560","1625","26"],["1708","63910","1123122","12321400","94621800","540668310","2385633250","8328360040","23388264900","53449696300","100216296180","155042514900","198684606120","211415508150","187022074050","137545109520","83996066700","42477615180","17708590900","6044338300","1672310640","369719350","63983010","8408400","800800","51142","1890","28"]],"degrees":[-1,1,3,5,7,9,11,13,15,17,19,21,23,25,27],"row_sums":["0","7","120","1275","11000","84375","600000","4046875","26250000","165234375","1015625000","6123046875","36328125000","212646484375","1230468750000"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["5","2"]],"recurrence":{"denominator":{"coeffs":[["13","7","-3","-2"],["-22","-19","-1","2"],["9","12","4","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["-4","-7","18","28","13","2"],["9","39","49","24","4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180981,"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-f141dbf83e78bcc3"]},{"id":180982,"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-f141dbf83e78bcc3"]},{"id":180983,"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-f141dbf83e78bcc3"]}],"region_exact":{},"incoming_edges":[{"id":"e-f141dbf83e78bcc3","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-f3c39411cf79b61b","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":"429a2992871f3d1997792d8b9e60349dbade65a206bbb69e741e75ec1905ff84","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-b60ac0f95ba69845","role":"sequence","name":"derivative of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k=0 and 0<=k<=2n.","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: 5, 2;\n 2: 28, 58, 30, 4;\n 3: 69, 330, 495, 300, 75, 6;\n 4: 128, 980, 2772, 3660, 2460, 852, 140, 8;\n 5: 205, 2170, 9210, 20200, 24915, 17970, 7630, 1840, 225, 10;\n 6: 300, 4062, 23130, 72240, 135960, 160770, 121926, 59520, 18360, 3390, 330, 12;\n 7: 413, 6818, 48825, 199164, 511980, 869484, 999229, 786520, 424872, 155820, 37653, 5628, 455, 14;\n 8: 544, 10600, 91560, 462280, 1516200, 3401328, 5364800, 6040280, 4892040, 2853200, 1190112, 348600, 69160, 8680, 600, 16;\n 9: 693, 15570, 157572, 949680, 3799530, 10657836, 21609000, 32233824, 35738415, 29603790, 18332424, 8447040, 2861586, 696780, 117180, 12672, 765, 18;\n","recurrence_sentence":"The row polynomials satisfy (9*n^2-22*n+13+(12*n^2-19*n+7)*x+(4*n^2-n-3)*x^2+(2*n-2)*x^3)*P(n,x) = (9*n^2-4*n+(39*n^2-7*n)*x+(49*n^2+18*n)*x^2+(24*n^2+28*n)*x^3+(4*n^2+13*n)*x^4+2*n*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5 + 2*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 5 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^2 - 4*n + (39*n^2 - 7*n)*x + (49*n^2 + 18*n)*x^2 + (24*n^2 + 28*n)*x^3 + (4*n^2 + 13*n)*x^4 + 2*n*x^5)*P[n - 1, x])/(9*n^2 - 22*n + 13 + (12*n^2 - 19*n + 7)*x + (4*n^2 - n - 3)*x^2 + (2*n - 2)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^2 - 4*n + (39*n^2 - 7*n)*x + (49*n^2 + 18*n)*x^2 + (24*n^2 + 28*n)*x^3 + (4*n^2 + 13*n)*x^4 + 2*n*x^5)*v[i - 1])/(9*n^2 - 22*n + 13 + (12*n^2 - 19*n + 7)*x + (4*n^2 - n - 3)*x^2 + (2*n - 2)*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-b30e7a5b4198051a","steps":["D","D+1"],"edges":["e-5fb206b09638d5b7","e-f141dbf83e78bcc3"],"text":"A272866 \u2192 D \u2192 D+1"},"headline":"A272866 \u2192 D \u2192 D+1","chain":null}