{"id":"n-953a0cb7db025175","revision":0,"first_index":0,"fingerprint":"f6-f42e153acb811e0d","name":"derivative of basis_map_A013610 of Unsigned 2-Stirling numbers of the first kind.","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":[["21","18"],["141","216","81"],["1026","2142","1458","324"],["8262","21150","19845","8100","1215"],["73656","218772","253935","144180","40095","4374"],["723384","2410884","3269889","2313360","901530","183708","15309"],["7778160","28449144","43547868","36213156","17690400","5082588","796068","52488"],["91002960","359609544","607047696","572536188","330393735","119593908","26545806","3306744","177147"],["1151910720","4861332288","8902815300","9298625040","6111735525","2624544342","737128350","130695120","13286025","590490"],["15693341760","70108784064","137612594196","156493386720","114322405725","56121324336","18768573747","4226333760","613814355","51963120","1948617"],["229047315840","1075683027456","2242065847032","2741357790864","2187363880350","1197259201314","459184667634","123518068344","22857277410","2773531530","198758934","6377292"],["3566477174400","17509529306880","38471561329032","50088630251088","43090549920990","25833397988970","11078177219109","3427118457048","759732641811","117793897650","12134038059","746143164","20726199"],["59025146342400","301551331956480","694337863145472","955304329333536","877281162177420","568460347539084","267667728467139","92816217206856","23722247061513","4417144501770","582864471189","51649687908","2756584467","66961566"],["1034811672652800","5480523521314560","13160724667277184","19017525051252000","18496170992016900","12822038092003140","6539962600328157","2495593618689600","716532933118860","154165613401800","24485272866054","2785601145600","214723421640","10044234900","215233605"],["19160141249894400","104858293419578880","261555399856904832","394958349591644736","404246796798499200","297382901227317360","162596460127917816","67343414051823624","21320221259216880","5163424151628600","949481435814912","130274440909704","12918225312720","874518051960","36159245640","688747536"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["39","438","4950","58572","735012","9818064","139609872","2110213728","33832663200","574014470400","10281315744000","193955869094400","3845220855436800","79945544618956800","1739678987202508800"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["21","18"],["141","216","81"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-9","-18","-9"],["-6","-6","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["7","6"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189559,"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-6c2e03a90b20db5f"]},{"id":189560,"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-6c2e03a90b20db5f"]},{"id":189561,"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-6c2e03a90b20db5f"]},{"id":189562,"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-6c2e03a90b20db5f"]}],"region_exact":{},"incoming_edges":[{"id":"e-6c2e03a90b20db5f","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-953a0cb7db025175","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":"fc17b3fa90a9a20de8c11c9fafaf49fc08b4553415b8bfda4ef614a8c00faf01","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-0d23d1c7119dd8ef","role":"sequence","name":"basis_map_A013610 of Unsigned 2-Stirling numbers of the first kind.","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-5fa2b7df4e1697ec","kind":"fingerprint_related","node_a_id":"n-871aaae0760b316f","node_b_id":"n-953a0cb7db025175","fingerprint":"f6-f42e153acb811e0d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f42e153acb811e0d"},"other_id":"n-871aaae0760b316f","other":{"id":"n-871aaae0760b316f","name":"derivative of basis_map_A013610 of Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","first_index":0,"fingerprint":"f6-f42e153acb811e0d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["3"],["21","18"]],"card_rows":[["3"],["21","18"],["141","216","81"],["1026","2142","1458","324"]],"row_sums_preview":["0","3","39","438","4950","58572","735012","9818064"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-914700b1060feeb2","steps":["basis A013610","D"],"edges":["e-8500cb477569290a","e-a58e8701216e3e4f"],"text":"A136124 \u2192 basis A013610 \u2192 D"},"headline":"A136124 \u2192 basis A013610 \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:          21,          18;\n  1:         141,         216,           81;\n  2:        1026,        2142,         1458,          324;\n  3:        8262,       21150,        19845,         8100,         1215;\n  4:       73656,      218772,       253935,       144180,        40095,        4374;\n  5:      723384,     2410884,      3269889,      2313360,       901530,      183708,       15309;\n  6:     7778160,    28449144,     43547868,     36213156,     17690400,     5082588,      796068,      52488;\n  7:    91002960,   359609544,    607047696,    572536188,    330393735,   119593908,    26545806,    3306744,    177147;\n  8:  1151910720,  4861332288,   8902815300,   9298625040,   6111735525,  2624544342,   737128350,  130695120,  13286025,   590490;\n  9: 15693341760, 70108784064, 137612594196, 156493386720, 114322405725, 56121324336, 18768573747, 4226333760, 613814355, 51963120, 1948617;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(6*n+18)*x+9*x^2)*P(n-2,x) + (2*n+7+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 21 + 18*x, P(1,x) = 141 + 216*x + 81*x^2.","mathematica":"P[0, x_] := 21 + 18*x;\nP[1, x_] := 141 + 216*x + 81*x^2;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (6*n + 18)*x + 9*x^2)*P[n - 2, x] + (2*n + 7 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 21 + 18*x; v[2] = 141 + 216*x + 81*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (6*n + 18)*x + 9*x^2)*v[i - 2] + (2*n + 7 + 6*x)*v[i - 1]); 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-c237629cf5fdbb07","steps":["basis A013610","D"],"edges":["e-36015c56c1237184","e-6c2e03a90b20db5f"],"text":"A143491 \u2192 basis A013610 \u2192 D"},"headline":"A143491 \u2192 basis A013610 \u2192 D","chain":null}
