{"id":"n-415ba9adacda5f2f","revision":0,"first_index":1,"fingerprint":"f6-563234e7acaf8cdf","name":"derivative of Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x)^4, read by rows.","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":[["6","2"],["15","30","3"],["34","180","102","4"],["73","812","1218","292","5"],["152","3166","9744","6332","760","6"],["311","11322","62721","83628","28305","1866","7"],["630","38320","352782","836280","587970","114960","4410","8"],["1269","125040","1813488","7003656","8754570","3626976","437640","10152","9"],["2548","397962","8754432","51944168","105054840","77916252","20427008","1591848","22932","10"],["5107","1244846","40390215","353101448","1084840510","1301808612","617927534","107707240","5601807","51070","11"],["10226","3846892","180233550","2250189820","10040057540","18225320568","14056080556","4500379640","540700650","19234460","112486","12"],["20465","11786100","784407918","13654063100","85554980715","223961796360","261288762420","136887969144","30721641975","2614693060","64823550","245580","13"],["40944","35890390","3349781472","79774964964","684005673040","2491726366242","4180620198720","3322301821656","1231210211472","199437412410","12282532064","215342340","532272","14"],["81903","108817602","14098988493","452471328372","5200939806495","25650152640174","59608088435301","68123529640344","38475228960261","10401879612990","1244296153023","56395953972","707314413","1146642","15"],["163822","328909896","58681123614","2506739109072","37992708764550","248203473926952","776116488284438","1226223533526192","997864056365706","413672456544920","83583959282010","7520217327216","254284868994","2302369272","2457330","16"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["8","48","320","2400","20160","188160","1935360","21772800","266112000","3512678400","49816166400","755545190400","12204960768000","209227898880000","3793999233024000"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["6","2"],["15","30","3"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","2","0"],["-2","0","1"],["0","-2","0"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1","1","0"],["0","-1","0","1"],["0","0","-1","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","0"],["3","0"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185382,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-3b0ae52d8be31782"]},{"id":185383,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-3b0ae52d8be31782"]},{"id":185384,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-3b0ae52d8be31782"]},{"id":185385,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-3b0ae52d8be31782"]}],"region_exact":{},"incoming_edges":[{"id":"e-3b0ae52d8be31782","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-415ba9adacda5f2f","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":"67b12b309f582be5d2163b8edba4819c23971929f1d75409748ecc1ed9cf6594","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9a3eef8528def021","role":"sequence","name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x)^4, read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    6,       2;\n   2:   15,      30,        3;\n   3:   34,     180,      102,         4;\n   4:   73,     812,     1218,       292,          5;\n   5:  152,    3166,     9744,      6332,        760,          6;\n   6:  311,   11322,    62721,     83628,      28305,       1866,         7;\n   7:  630,   38320,   352782,    836280,     587970,     114960,      4410,         8;\n   8: 1269,  125040,  1813488,   7003656,    8754570,    3626976,    437640,     10152,       9;\n   9: 2548,  397962,  8754432,  51944168,  105054840,   77916252,  20427008,   1591848,   22932,    10;\n  10: 5107, 1244846, 40390215, 353101448, 1084840510, 1301808612, 617927534, 107707240, 5601807, 51070, 11;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(2*n+2+(2*n^2-2)*x+(n^3-n)*x^2)*P(n-2,x) - ((n+1)*x+(n^2-1)*x^2-(n^2+n)*x^3)*d/dx P(n-2,x) + (3*n+1+2*n^2*x)*P(n-1,x) + (n*x-n*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 6 + 2*x, P(2,x) = 15 + 30*x + 3*x^2.","mathematica":"P[1, x_] := 6 + 2*x;\nP[2, x_] := 15 + 30*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 2 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*P[n - 2, x] - ((n + 1)*x + (n^2 - 1)*x^2 - (n^2 + n)*x^3)*D[P[n - 2, x], x] + (3*n + 1 + 2*n^2*x)*P[n - 1, x] + (n*x - n*x^2)*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 6 + 2*x; v[2] = 15 + 30*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = (-(2*n + 2 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*v[i - 2] - ((n + 1)*x + (n^2 - 1)*x^2 - (n^2 + n)*x^3)*deriv(v[i - 2], x) + (3*n + 1 + 2*n^2*x)*v[i - 1] + (n*x - n*x^2)*deriv(v[i - 1], x))/(n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-9a3eef8528def021","steps":["D"],"edges":["e-3b0ae52d8be31782"],"text":"A166344 \u2192 D"},"headline":"A166344 \u2192 D","chain":null}