{"id":"n-3f9e261fec4bd38f","revision":0,"first_index":0,"fingerprint":"f6-d87433d07b949f42","name":"derivative of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.","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":[[],["2"],["32","8"],["312","624","24"],["2656","17472","7968","64"],["21664","350848","526272","86656","160"],["174336","6038400","21050880","12076800","871680","384"],["1397120","95695104","656515200","875353600","239237760","8382720","896"],["11182592","1445912576","17723668992","45518387200","29539448320","4337737728","78278144","2048"],["89473536","21226844160","436890396672","1939687944192","2424609930240","873780793344","74293954560","715788288","4608"],["715816960","306123171840","10138779648000","72567666647040","155175035535360","108851499970560","23657152512000","1224492687360","6442352640","10240"],["5726599168","4365895905280","225734830848000","2481567738839040","8412565302865920","10095078363439104","4342743542968320","601959548928000","19646531573760","57265991680","22528"],["45812932608","61832640970752","4881431873003520","79569749933015040","405688315327856640","767225955621371904","567963641458999296","159139499866030080","14644295619010560","309163204853760","503942258688","49152"],["366503763968","871887532425216","103379073070350336","2433293744775987200","17955135461304852480","50574929362922962944","59004084256743456768","28728216738087763968","5474910925745971200","344596910234501120","4795381428338688","4398045167616","106496"],["2932030767104","12260668011020288","2156513989714378752","71811640737991294976","744894896896074383360","2999155443469363740672","5192359414593424195584","3998873924625818320896","1340810814412933890048","179529101844978237440","7907217962286055424","73564008066121728","38116399972352","229376"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","40","960","28160","985600","40212480","1876582400","98654617600","5771295129600","371927908352000","26183724747980800","1999484435300352000","164624218506395648000","14537584834257092608000"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-5"],["3"]]},"terms":[{"coeff":{"coeffs":[["32","-128","320"],["-48","240","-792"],["0","-72","540"],["0","0","-108"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","24","-144","120"],["0","-36","288","-252"],["0","0","-108","108"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-44","100"],["30","-120"],["0","36"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-30","30"],["0","18","-18"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191195,"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-c27fe42c4f5f405b"]},{"id":191196,"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-c27fe42c4f5f405b"]},{"id":191197,"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-c27fe42c4f5f405b"]},{"id":191198,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-c27fe42c4f5f405b"]}],"region_exact":{},"incoming_edges":[{"id":"e-c27fe42c4f5f405b","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-3f9e261fec4bd38f","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":"b8502a06083c126deaf2afdc828a55fd0875d60efff439c216752f7d79dd694e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c060a08f0b5770ad","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.","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":[{"id":"l-bd50c5b2f79379ac","kind":"fingerprint_related","node_a_id":"n-f1df042cd3a7a7cc","node_b_id":"n-3f9e261fec4bd38f","fingerprint":"f6-d87433d07b949f42","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d87433d07b949f42"},"other_id":"n-f1df042cd3a7a7cc","other":{"id":"n-f1df042cd3a7a7cc","name":"derivative of Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).","first_index":0,"fingerprint":"f6-d87433d07b949f42","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["8","2"]],"card_rows":[["1"],["8","2"],["39","78","3"],["166","1092","498","4"]],"row_sums_preview":["0","1","10","120","1760","30800","628320","14660800"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ddb0cd3a4d6edf09","steps":["D"],"edges":["e-728e3d5cc0dcc97e"],"text":"A142458 \u2192 D"},"headline":"A142458 \u2192 D","chain":null}}],"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:        2;\n  2:       32,           8;\n  3:      312,         624,           24;\n  4:     2656,       17472,         7968,            64;\n  5:    21664,      350848,       526272,         86656,           160;\n  6:   174336,     6038400,     21050880,      12076800,        871680,          384;\n  7:  1397120,    95695104,    656515200,     875353600,     239237760,      8382720,         896;\n  8: 11182592,  1445912576,  17723668992,   45518387200,   29539448320,   4337737728,    78278144,      2048;\n  9: 89473536, 21226844160, 436890396672, 1939687944192, 2424609930240, 873780793344, 74293954560, 715788288, 4608;\n","recurrence_sentence":"The row polynomials satisfy (3*n-5)*P(n,x) = -(48*n-32+(72*n^2-240*n+128)*x+(108*n^3-540*n^2+792*n-320)*x^2)*P(n-2,x) - ((36*n-24)*x+(108*n^2-288*n+144)*x^2-(108*n^2-252*n+120)*x^3)*d/dx P(n-2,x) + (30*n-44+(36*n^2-120*n+100)*x)*P(n-1,x) + ((18*n-30)*x-(18*n-30)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(48*n - 32 + (72*n^2 - 240*n + 128)*x + (108*n^3 - 540*n^2 + 792*n - 320)*x^2)*P[n - 2, x] - ((36*n - 24)*x + (108*n^2 - 288*n + 144)*x^2 - (108*n^2 - 252*n + 120)*x^3)*D[P[n - 2, x], x] + (30*n - 44 + (36*n^2 - 120*n + 100)*x)*P[n - 1, x] + ((18*n - 30)*x - (18*n - 30)*x^2)*D[P[n - 1, x], x])/(3*n - 5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(48*n - 32 + (72*n^2 - 240*n + 128)*x + (108*n^3 - 540*n^2 + 792*n - 320)*x^2)*v[i - 2] - ((36*n - 24)*x + (108*n^2 - 288*n + 144)*x^2 - (108*n^2 - 252*n + 120)*x^3)*deriv(v[i - 2], x) + (30*n - 44 + (36*n^2 - 120*n + 100)*x)*v[i - 1] + ((18*n - 30)*x - (18*n - 30)*x^2)*deriv(v[i - 1], x))/(3*n - 5)); 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-c060a08f0b5770ad","steps":["D"],"edges":["e-c27fe42c4f5f405b"],"text":"A257616 \u2192 D"},"headline":"A257616 \u2192 D","chain":null}
