{"id":"n-fc94df5213f0c4e4","revision":0,"first_index":1,"fingerprint":"f6-b50e0bffdfcf6785","name":"theta_plus_one_squared of reversal of Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).","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"],["2","4"],["3","24","9"],["4","72","126","16"],["5","160","675","480","25"],["6","300","2340","4320","1550","36"],["7","504","6300","22400","22575","4536","49"],["8","784","14364","84000","170100","104328","12446","64"],["9","1152","29106","254016","868875","1118880","444675","32640","81"],["10","1620","54000","658560","3424050","7654500","6684580","1791360","82782","100"],["11","2200","93555","1520640","11197725","38769192","60448850","37312000","6925743","204600","121"],["12","2904","153450","3207600","31805400","158103792","389153688","441408000","198126324","25957800","495374","144"],["13","3744","240669","6292000","80887950","547088256","1960390432","3579383808","3040942905","1013012000","94969875","1179360","169"],["14","4732","363636","11627616","188188000","1663782120","8197629440","22099517440","30830077278","20037517500","5029604580","340835040","2768558","196"],["15","5880","532350","20442240","406856450","4558554000","29603664090","110913454080","231756335811","252415963800","127513986600","24397027200","1204176207","6422136","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","36","218","1345","8552","56371","386094","2749434","20351562","156474637","1248414488","10324391181","88401916150","782797421084"],"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"],["2","4"],["3","24","9"],["4","72","126","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","20","190"],["0","-4","-83"],["0","0","9"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","-16","140"],["0","0","4","-50"],["0","0","0","4"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","17"],["0","0","0","0","-4"]]},"deriv_order":2,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["4","-44","-122"],["0","12","103"],["0","0","-18"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","32","-113"],["0","0","-12","72"],["0","0","0","-8"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","-18"],["0","0","0","0","8"]]},"deriv_order":2,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-6","28","4"],["0","-12","-20"],["0","0","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-16","5"],["0","0","12","-22"],["0","0","0","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","1"],["0","0","0","0","-4"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","-4"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177012,"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-9f79bfd2d08179a6"]},{"id":177013,"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-9f79bfd2d08179a6"]},{"id":177014,"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-9f79bfd2d08179a6"]}],"region_exact":{},"incoming_edges":[{"id":"e-9f79bfd2d08179a6","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-fc94df5213f0c4e4","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":"bd001ca861d48d211d58f778c92515499436eed4b3a768a3322f4e679f8d46b3","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-726472edd5bb65cd","role":"sequence","name":"reversal of Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).","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:  2,    4;\n   3:  3,   24,     9;\n   4:  4,   72,   126,     16;\n   5:  5,  160,   675,    480,      25;\n   6:  6,  300,  2340,   4320,    1550,      36;\n   7:  7,  504,  6300,  22400,   22575,    4536,      49;\n   8:  8,  784, 14364,  84000,  170100,  104328,   12446,      64;\n   9:  9, 1152, 29106, 254016,  868875, 1118880,  444675,   32640,    81;\n  10: 10, 1620, 54000, 658560, 3424050, 7654500, 6684580, 1791360, 82782, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+(4*n-20)*x-(9*n^2-83*n+190)*x^2)*P(n-4,x) + ((4*n-16)*x^2+(4*n^2-50*n+140)*x^3)*d/dx P(n-4,x) - (4*n-17)*x^4*d^2/dx^2 P(n-4,x) + (4+(12*n-44)*x-(18*n^2-103*n+122)*x^2)*P(n-3,x) - ((12*n-32)*x^2+(8*n^2-72*n+113)*x^3)*d/dx P(n-3,x) + (8*n-18)*x^4*d^2/dx^2 P(n-3,x) - (6+(12*n-28)*x-(9*n^2-20*n+4)*x^2)*P(n-2,x) + ((12*n-16)*x^2+(4*n^2-22*n+5)*x^3)*d/dx P(n-2,x) - (4*n-1)*x^4*d^2/dx^2 P(n-2,x) + (4+(4*n-4)*x)*P(n-1,x) - 4*n*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2 + 4*x, P(3,x) = 3 + 24*x + 9*x^2, P(4,x) = 4 + 72*x + 126*x^2 + 16*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 4*x;\nP[3, x_] := 3 + 24*x + 9*x^2;\nP[4, x_] := 4 + 72*x + 126*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[-(1 + (4*n - 20)*x - (9*n^2 - 83*n + 190)*x^2)*P[n - 4, x] + ((4*n - 16)*x^2 + (4*n^2 - 50*n + 140)*x^3)*D[P[n - 4, x], x] - (4*n - 17)*x^4*D[P[n - 4, x], {x, 2}] + (4 + (12*n - 44)*x - (18*n^2 - 103*n + 122)*x^2)*P[n - 3, x] - ((12*n - 32)*x^2 + (8*n^2 - 72*n + 113)*x^3)*D[P[n - 3, x], x] + (8*n - 18)*x^4*D[P[n - 3, x], {x, 2}] - (6 + (12*n - 28)*x - (9*n^2 - 20*n + 4)*x^2)*P[n - 2, x] + ((12*n - 16)*x^2 + (4*n^2 - 22*n + 5)*x^3)*D[P[n - 2, x], x] - (4*n - 1)*x^4*D[P[n - 2, x], {x, 2}] + (4 + (4*n - 4)*x)*P[n - 1, x] - 4*n*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] = 2 + 4*x; v[3] = 3 + 24*x + 9*x^2; v[4] = 4 + 72*x + 126*x^2 + 16*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + (4*n - 20)*x - (9*n^2 - 83*n + 190)*x^2)*v[i - 4] + ((4*n - 16)*x^2 + (4*n^2 - 50*n + 140)*x^3)*deriv(v[i - 4], x) - (4*n - 17)*x^4*deriv(deriv(v[i - 4], x), x) + (4 + (12*n - 44)*x - (18*n^2 - 103*n + 122)*x^2)*v[i - 3] - ((12*n - 32)*x^2 + (8*n^2 - 72*n + 113)*x^3)*deriv(v[i - 3], x) + (8*n - 18)*x^4*deriv(deriv(v[i - 3], x), x) - (6 + (12*n - 28)*x - (9*n^2 - 20*n + 4)*x^2)*v[i - 2] + ((12*n - 16)*x^2 + (4*n^2 - 22*n + 5)*x^3)*deriv(v[i - 2], x) - (4*n - 1)*x^4*deriv(deriv(v[i - 2], x), x) + (4 + (4*n - 4)*x)*v[i - 1] - 4*n*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-3cb22164e2f83fe1","steps":["\u03b8+1","reverse","(\u03b8+1)\u00b2"],"edges":["e-907e3c6c6275331d","e-6878e8dc6c11eb74","e-9f79bfd2d08179a6"],"text":"A008277 \u2192 \u03b8+1 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A008277 \u2192 \u03b8+1 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}