{"id":"n-be6f49875ee2b6af","revision":0,"first_index":1,"fingerprint":"f6-c70146f291298cbd","name":"theta_plus_one_squared of Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).","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":[["1"],["1","8"],["1","24","54"],["1","56","324","384"],["1","120","1350","3840","3000"],["1","248","4860","24960","45000","25920"],["1","504","16254","134400","420000","544320","246960"],["1","1016","52164","653184","3150000","6894720","6914880","2580480"],["1","2040","163350","2983680","20853000","68584320","114095520","92897280","29393280"],["1","4088","503820","13096320","127575000","591675840","1452124800","1935360000","1322697600","362880000"],["1","8184","1539054","55968000","740190000","4652303040","15802229520","30656102400","33949238400","19958400000","4829932800"],["1","16376","4672404","234816384","4138200000","34309059840","154941716160","410365992960","654735312000","618710400000","318775564800","68976230400"],["1","32760","14127750","972491520","22525503000","241608407040","1411481111040","4901910773760","10566942946560","14270256000000","11741566636800","5380145971200","1052366515200"],["1","65528","42604380","3990430080","120225105000","1644270788160","12182358988800","53963823513600","150938313926400","273158645760000","319094340364800","232242967756800","95765352883200","17086945075200"],["1","131064","128255454","16264684800","632300760000","10904369636160","100942759597680","559003808563200","1973126502547200","4595022432000000","7145779319078400","7343899250688000","4788267644160000","1794129232896000","294226732800000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","79","765","8311","100989","1362439","20246445","328972471","5805917469","110645911399","2265191981325","49589790516631","1156201277261949","28605950745797959"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-8","22"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-2","0","26"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","4","5"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182938,"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-ca96520d8e40bf30"]},{"id":182939,"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-ca96520d8e40bf30"]},{"id":182940,"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-ca96520d8e40bf30"]}],"region_exact":{},"incoming_edges":[{"id":"e-ca96520d8e40bf30","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-be6f49875ee2b6af","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"753627d8c4d37a402a5cbd1c056e886b7ac4ff1511e0d304589d2fbc7d583b6c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d938c1937a93c4df","role":"sequence","name":"Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).","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: 1;\n   2: 1,    8;\n   3: 1,   24,     54;\n   4: 1,   56,    324,      384;\n   5: 1,  120,   1350,     3840,      3000;\n   6: 1,  248,   4860,    24960,     45000,     25920;\n   7: 1,  504,  16254,   134400,    420000,    544320,     246960;\n   8: 1, 1016,  52164,   653184,   3150000,   6894720,    6914880,    2580480;\n   9: 1, 2040, 163350,  2983680,  20853000,  68584320,  114095520,   92897280,   29393280;\n  10: 1, 4088, 503820, 13096320, 127575000, 591675840, 1452124800, 1935360000, 1322697600, 362880000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(8*x-22*x^2)*P(n-2,x) - (2*x-26*x^3)*d/dx P(n-2,x) - (x^2-4*x^3-5*x^4)*d^2/dx^2 P(n-2,x) + (1+8*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 8*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-(8*x - 22*x^2)*P[n - 2, x] - (2*x - 26*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3 - 5*x^4)*D[P[n - 2, x], {x, 2}] + (1 + 8*x)*P[n - 1, x] + (2*x - 4*x^2)*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] = 1; v[2] = 1 + 8*x; for(n = 3, N, my(i = n); v[i] = -(8*x - 22*x^2)*v[i - 2] - (2*x - 26*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3 - 5*x^4)*deriv(deriv(v[i - 2], x), x) + (1 + 8*x)*v[i - 1] + (2*x - 4*x^2)*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-d938c1937a93c4df","steps":["(\u03b8+1)\u00b2"],"edges":["e-ca96520d8e40bf30"],"text":"A019538 \u2192 (\u03b8+1)\u00b2"},"headline":"A019538 \u2192 (\u03b8+1)\u00b2","chain":null}