{"id":"n-63d34f3b7f5dc912","revision":0,"first_index":1,"fingerprint":"f6-1216106ea1ae9807","name":"one_plus_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":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["7","8","1"],["16","45","18","1"],["35","214","192","38","1"],["74","885","1624","698","78","1"],["153","3318","11327","9580","2343","158","1"],["312","11633","68382","104535","49212","7527","318","1"],["631","38950","371942","953874","797040","232554","23570","638","1"],["1270","126309","1876008","7608152","10505484","5377890","1042136","72672","1278","1"],["2549","400510","8953413","54862312","118040882","98927220","33413050","4509992","221913","2558","1"],["5108","1249953","41012638","366564853","1173115872","1518776714","834895636","195982602","19065212","673493","5118","1"],["10227","3857118","182156996","2310267670","10602604995","20233332076","17093633984","6508391148","1103248105","79312310","2035932","10238","1"],["20466","11806565","790300968","13915532406","88968496490","241072792503","298615728480","174214935204","47832638118","6028208835","326292856","6138630","20478","1"],["40945","35931334","3367726667","80891558788","703949414281","2628527500850","4595907926427","3919533278616","1646497939179","336238547018","32226273305","1331936164","18477467","40958","1"],["81904","108899505","14153397294","457170991203","5314057638588","26690340601473","63883113875330","76638970845387","46990670165304","14676905053019","2284484114322","169513786065","5406977244","55555443","81918","1"],["163823","329073718","58845578562","2526299483610","38619393541818","255802015679862","817483733938930","1337097317566826","1151141998056480","524546240585554","124951204936502","15118759080126","880969646262","21862743810","166912278","163838","1"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["16","80","480","3360","26880","241920","2419200","26611200","319334400","4151347200","58118860800","871782912000","13948526592000","237124952064000","4268249137152000"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["7","8","1"],["16","45","18","1"]],"recurrence":{"denominator":{"coeffs":[["-1","2"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["0","-4","6","-2","0"],["-2","3","-5","4","-2"],["0","-2","2","-3","0"],["0","0","-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2","4","-4","2"],["0","-1","2","-4","3","0"],["0","0","-1","1","0","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","7","-4","2"],["3","-3","5","0"],["0","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","3","-2"],["0","1","-1","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185386,"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-386fbbda9a5e739f"]},{"id":185387,"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-386fbbda9a5e739f"]},{"id":185388,"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-386fbbda9a5e739f"]},{"id":185389,"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-386fbbda9a5e739f"]}],"region_exact":{},"incoming_edges":[{"id":"e-386fbbda9a5e739f","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-63d34f3b7f5dc912","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":"b7a997530f6762be4339c0b644c9be4393bccb9fe56af9b446f3877a0cb23fae","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:    7,       8,        1;\n   2:   16,      45,       18,         1;\n   3:   35,     214,      192,        38,          1;\n   4:   74,     885,     1624,       698,         78,          1;\n   5:  153,    3318,    11327,      9580,       2343,        158,         1;\n   6:  312,   11633,    68382,    104535,      49212,       7527,       318,         1;\n   7:  631,   38950,   371942,    953874,     797040,     232554,     23570,       638,        1;\n   8: 1270,  126309,  1876008,   7608152,   10505484,    5377890,   1042136,     72672,     1278,      1;\n   9: 2549,  400510,  8953413,  54862312,  118040882,   98927220,  33413050,   4509992,   221913,   2558,    1;\n  10: 5108, 1249953, 41012638, 366564853, 1173115872, 1518776714, 834895636, 195982602, 19065212, 673493, 5118, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1+2*x)*P(n,x) = -(2*n+(2*n^2-3*n+4)*x+(n^3-2*n^2+5*n-6)*x^2+(3*n^2-4*n+2)*x^3+2*n*x^4)*P(n-2,x) - (n*x+(n^2-2*n+2)*x^2-(n^2-4*n+4)*x^3-(3*n-4)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n-2+(2*n^2-3*n+7)*x+(5*n-4)*x^2+2*x^3)*P(n-1,x) + ((n-1)*x-(n-3)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 7 + 8*x + x^2, P(2,x) = 16 + 45*x + 18*x^2 + x^3.","mathematica":"P[1, x_] := 7 + 8*x + x^2;\nP[2, x_] := 16 + 45*x + 18*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + (2*n^2 - 3*n + 4)*x + (n^3 - 2*n^2 + 5*n - 6)*x^2 + (3*n^2 - 4*n + 2)*x^3 + 2*n*x^4)*P[n - 2, x] - (n*x + (n^2 - 2*n + 2)*x^2 - (n^2 - 4*n + 4)*x^3 - (3*n - 4)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n - 2 + (2*n^2 - 3*n + 7)*x + (5*n - 4)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 1)*x - (n - 3)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 1 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 7 + 8*x + x^2; v[2] = 16 + 45*x + 18*x^2 + x^3; for(n = 3, N, my(i = n); v[i] = (-(2*n + (2*n^2 - 3*n + 4)*x + (n^3 - 2*n^2 + 5*n - 6)*x^2 + (3*n^2 - 4*n + 2)*x^3 + 2*n*x^4)*v[i - 2] - (n*x + (n^2 - 2*n + 2)*x^2 - (n^2 - 4*n + 4)*x^3 - (3*n - 4)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n - 2 + (2*n^2 - 3*n + 7)*x + (5*n - 4)*x^2 + 2*x^3)*v[i - 1] + ((n - 1)*x - (n - 3)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 1 + 2*x)); 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+1"],"edges":["e-386fbbda9a5e739f"],"text":"A166344 \u2192 D+1"},"headline":"A166344 \u2192 D+1","chain":null}