{"id":"n-500e32224f9dbdfe","revision":0,"first_index":1,"fingerprint":"f6-dd3f1b3bb23145ea","name":"derivative of Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m = 0.","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":[["11","2"],["111","60","3"],["1111","1362","174","4"],["11111","28100","6915","380","5"],["111111","556122","235770","23280","705","6"],["1111111","10788540","7435743","1175720","61530","1176","7"],["11111111","207204482","224383614","53415964","4300030","138516","1820","8"],["111111111","3959107380","6593547915","2275568820","264571335","12778416","278082","2664","9"],["1111111111","75445262442","190558002690","92987443560","15014742435","1020298482","32705400","512280","3735","10"],["11111111111","1435682208620","5448791968983","3694612748640","806912456460","74134107432","3283493829","74774040","882585","5060","11"],["111111111111","27300184186002","154719698444454","143965727658324","41736238932960","5045670856512","296633397060","9211069296","156492765","1441110","6666","12"],["1111111111111","518925721756260","4373101832723715","5533024854617260","2099824150489065","327595383827712","24871153411104","1011417655248","23194859688","305021860","2251821","8580","13"],["11111111111111","9861810935591162","123225239898898410","210552722064470240","103508191990768565","20537535091111038","1973948432776320","102257664160080","3039823356570","53529055580","560706146","3391752","10829","14"],["111111111111111","187396629998454300","3465099433572542223","7954751036250596760","5024567734155941790","1253774260400029368","150293157303980691","9720750549715920","364305387418830","8248725565080","114952575738","981380400","4952220","13440","15"],["1111111111111111","3560758192192853922","97303879085028863694","298945920919368803084","241073554566486568290","74987065602988745388","11081498704588131820","881378398476668664","40808886136774470","1155417790220980","20568855695388","232373273496","1647524060","7040040","16440","16"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["13","174","2651","46511","926994","20573827","500555545","13216965732","376170082145","11474603756771","373184633596272","12879489127733437","469775911105407817","18046086783394277886","727877151687052290863"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["11","2"],["111","60","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-10","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-9","-9"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["11","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","9"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191618,"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-a39ae501649f9059"]},{"id":191619,"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-a39ae501649f9059"]},{"id":191620,"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-a39ae501649f9059"]},{"id":191621,"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-a39ae501649f9059"]}],"region_exact":{},"incoming_edges":[{"id":"e-a39ae501649f9059","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-500e32224f9dbdfe","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":"45cb26f68aba170b7b2b86d97a8d1de653ada6e9e101db467fe3cb0c0da7c29a","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-2b620f031a59b07c","role":"sequence","name":"Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m = 0.","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:          11,             2;\n   2:         111,            60,             3;\n   3:        1111,          1362,           174,             4;\n   4:       11111,         28100,          6915,           380,            5;\n   5:      111111,        556122,        235770,         23280,          705,           6;\n   6:     1111111,      10788540,       7435743,       1175720,        61530,        1176,          7;\n   7:    11111111,     207204482,     224383614,      53415964,      4300030,      138516,       1820,        8;\n   8:   111111111,    3959107380,    6593547915,    2275568820,    264571335,    12778416,     278082,     2664,      9;\n   9:  1111111111,   75445262442,  190558002690,   92987443560,  15014742435,  1020298482,   32705400,   512280,   3735,   10;\n  10: 11111111111, 1435682208620, 5448791968983, 3694612748640, 806912456460, 74134107432, 3283493829, 74774040, 882585, 5060, 11;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(10+2*x+x^2)*P(n-2,x) - (9*x+9*x^2)*d/dx P(n-2,x) + (11+2*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 11 + 2*x, P(2,x) = 111 + 60*x + 3*x^2.","mathematica":"P[1, x_] := 11 + 2*x;\nP[2, x_] := 111 + 60*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(10 + 2*x + x^2)*P[n - 2, x] - (9*x + 9*x^2)*D[P[n - 2, x], x] + (11 + 2*x)*P[n - 1, x] + 9*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 11 + 2*x; v[2] = 111 + 60*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = -(10 + 2*x + x^2)*v[i - 2] - (9*x + 9*x^2)*deriv(v[i - 2], x) + (11 + 2*x)*v[i - 1] + 9*x*deriv(v[i - 1], 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-2b620f031a59b07c","steps":["D"],"edges":["e-a39ae501649f9059"],"text":"A166979 \u2192 D"},"headline":"A166979 \u2192 D","chain":null}