{"id":"n-600339540c4efc1c","revision":0,"first_index":0,"fingerprint":"f6-eb1d45955591e332","name":"theta_plus_one of basis_map_A008277 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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":[["1"],["1","2"],["2","8","3"],["6","36","27","4"],["24","192","216","64","5"],["120","1200","1800","800","125","6"],["720","8640","16200","9600","2250","216","7"],["5040","70560","158760","117600","36750","5292","343","8"],["40320","645120","1693440","1505280","588000","112896","10976","512","9"],["362880","6531840","19595520","20321280","9525600","2286144","296352","20736","729","10"],["3628800","72576000","244944000","290304000","158760000","45722880","7408800","691200","36450","1000","11"],["39916800","878169600","3293136000","4390848000","2744280000","922078080","179292960","20908800","1470150","60500","1331","12"],["479001600","11496038400","47421158400","70253568000","49397040000","18968463360","4303031040","602173440","52925400","2904000","95832","1728","13"],["6227020800","161902540800","728561433600","1187285299200","927566640000","400708788480","103887463680","16961218560","1788878520","122694000","5398536","146016","2197","14"],["87178291200","2440992153600","11899836748800","21155265331200","18180306144000","8726546949120","2545242860160","474914119680","58436698320","4809604800","264528264","9539712","215306","2744","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","13","73","501","4051","37633","394353","4596553","58941091","824073141","12470162233","202976401213","3535017524403","65573803186921"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1","0"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["0","-2","1"],["-1","0","1"],["0","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","1"],["0","0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180825,"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-ef6dedb3b88ad994"]},{"id":180826,"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-ef6dedb3b88ad994"]},{"id":180827,"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-ef6dedb3b88ad994"]},{"id":180828,"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-ef6dedb3b88ad994"]}],"region_exact":{},"incoming_edges":[{"id":"e-ef6dedb3b88ad994","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-600339540c4efc1c","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":"b478354c6531cce7dcbb795a749032269e33db77f7bf576bf933a841f71c57fc","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-8ab17db79dad1880","role":"sequence","name":"basis_map_A008277 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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-1fb424ea7780e0db","kind":"fingerprint_related","node_a_id":"n-600339540c4efc1c","node_b_id":"n-3202cf200924ca0a","fingerprint":"f6-eb1d45955591e332","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-eb1d45955591e332"},"other_id":"n-3202cf200924ca0a","other":{"id":"n-3202cf200924ca0a","name":"theta_plus_one_squared of basis_map_A008277 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","first_index":0,"fingerprint":"f6-eb1d45955591e332","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["6","24","9"]],"card_rows":[["1"],["2","4"],["6","24","9"],["24","144","108","16"]],"row_sums_preview":["1","6","39","292","2505","24306","263431","3154824"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["basis A008277","(\u03b8+1)\u00b2"],"edges":["e-31e8b29857f291e5","e-06f013a34dc61285"],"text":"A130534 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2"},"headline":"A130534 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2","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:      1;\n  1:      1,       2;\n  2:      2,       8,        3;\n  3:      6,      36,       27,        4;\n  4:     24,     192,      216,       64,       5;\n  5:    120,    1200,     1800,      800,     125,       6;\n  6:    720,    8640,    16200,     9600,    2250,     216,      7;\n  7:   5040,   70560,   158760,   117600,   36750,    5292,    343,     8;\n  8:  40320,  645120,  1693440,  1505280,  588000,  112896,  10976,   512,   9;\n  9: 362880, 6531840, 19595520, 20321280, 9525600, 2286144, 296352, 20736, 729, 10;\n","recurrence_sentence":"The row polynomials satisfy -(1-n*x)*P(n,x) = -(n-(n^2-2)*x-(n+1)*x^2)*P(n-1,x) - (x-(n+1)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - (n^2 - 2)*x - (n + 1)*x^2)*P[n - 1, x] - (x - (n + 1)*x^2)*D[P[n - 1, x], x])/(-1 + n*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(n - (n^2 - 2)*x - (n + 1)*x^2)*v[i - 1] - (x - (n + 1)*x^2)*deriv(v[i - 1], x))/(-1 + n*x)); 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-b9fbfb57d9269e2f","steps":["basis A008277","\u03b8+1"],"edges":["e-07cfebc08bea7e3c","e-ef6dedb3b88ad994"],"text":"A132393 \u2192 basis A008277 \u2192 \u03b8+1"},"headline":"A132393 \u2192 basis A008277 \u2192 \u03b8+1","chain":null}