{"id":"n-85696c1481362941","revision":0,"first_index":0,"fingerprint":"f6-c06bc352c6c59952","name":"theta_plus_one of Triangle of cubed binomial coefficients, T(n,k) = C(n,k)^3, read by rows.","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"],["1","16","3"],["1","54","81","4"],["1","128","648","256","5"],["1","250","3000","4000","625","6"],["1","432","10125","32000","16875","1296","7"],["1","686","27783","171500","214375","55566","2401","8"],["1","1024","65856","702464","1715000","1053696","153664","4096","9"],["1","1458","139968","2370816","10001880","12002256","4148928","373248","6561","10"],["1","2000","273375","6912000","46305000","96018048","64827000","13824000","820125","10000","11"],["1","2662","499125","17968500","179685000","591666768","690277896","287496000","40429125","1663750","14641","12"],["1","3456","862488","42592000","606436875","2980758528","5522223168","3974344704","1091586375","106480000","3162456","20736","13"],["1","4394","1423656","93574624","1827629375","12790481418","35371207872","40424237568","19185722127","3655258750","257330216","5694624","28561","14"],["1","5488","2260713","192914176","5015015005","48144144048","189567567189","323393900544","243729729243","80240240080","11033033011","578742528","9796423","38416","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","20","140","1038","7882","60736","472320","3695810","29045126","228991560","1809703480","14328470800","113612593200","901907386880"],"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":[["0","0"],["0","0"],["1","0"],["0","0"],["0","0"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["0","12","0"],["0","-30","0"],["1","20","10"],["0","0","-18"],["0","0","0"],["0","1","10"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","24","0"],["0","3","-15","0"],["0","0","0","30"],["0","0","6","-6"],["0","0","3","-15"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","6","6","0"],["0","0","0","0","0"],["0","0","0","10","10"],["0","0","0","6","6"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180452,"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-21dc18e24ad527f8"]},{"id":180453,"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-21dc18e24ad527f8"]},{"id":180454,"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-21dc18e24ad527f8"]}],"region_exact":{},"incoming_edges":[{"id":"e-21dc18e24ad527f8","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-85696c1481362941","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":"454a609f40a29b9e459ebbf79c7d6e84db840bc545d790e851a774f262f37f74","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-fabb0a68e001a863","role":"sequence","name":"Triangle of cubed binomial coefficients, T(n,k) = C(n,k)^3, read by rows.","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":[],"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: 1,   16,      3;\n  3: 1,   54,     81,       4;\n  4: 1,  128,    648,     256,        5;\n  5: 1,  250,   3000,    4000,      625,        6;\n  6: 1,  432,  10125,   32000,    16875,     1296,       7;\n  7: 1,  686,  27783,  171500,   214375,    55566,    2401,      8;\n  8: 1, 1024,  65856,  702464,  1715000,  1053696,  153664,   4096,    9;\n  9: 1, 1458, 139968, 2370816, 10001880, 12002256, 4148928, 373248, 6561, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n^5*x)*P(n,x) = (n^2+(n^5+20*n^2-30*n+12)*x+(10*n^5-18*n^3+10*n^2)*x^2)*P(n-1,x) + ((3*n+6)*x+(3*n^4+6*n^3-15*n+24)*x^2-(15*n^4+6*n^3-30*n^2)*x^3)*d/dx P(n-1,x) + (6*x^2+(6*n^3+10*n^2+6)*x^3+(6*n^3+10*n^2)*x^4)*d^2/dx^2 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^2 + (n^5 + 20*n^2 - 30*n + 12)*x + (10*n^5 - 18*n^3 + 10*n^2)*x^2)*P[n - 1, x] + ((3*n + 6)*x + (3*n^4 + 6*n^3 - 15*n + 24)*x^2 - (15*n^4 + 6*n^3 - 30*n^2)*x^3)*D[P[n - 1, x], x] + (6*x^2 + (6*n^3 + 10*n^2 + 6)*x^3 + (6*n^3 + 10*n^2)*x^4)*D[P[n - 1, x], {x, 2}])/(n^2 + n^5*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^2 + (n^5 + 20*n^2 - 30*n + 12)*x + (10*n^5 - 18*n^3 + 10*n^2)*x^2)*v[i - 1] + ((3*n + 6)*x + (3*n^4 + 6*n^3 - 15*n + 24)*x^2 - (15*n^4 + 6*n^3 - 30*n^2)*x^3)*deriv(v[i - 1], x) + (6*x^2 + (6*n^3 + 10*n^2 + 6)*x^3 + (6*n^3 + 10*n^2)*x^4)*deriv(deriv(v[i - 1], x), x))/(n^2 + n^5*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-fabb0a68e001a863","steps":["\u03b8+1"],"edges":["e-21dc18e24ad527f8"],"text":"A181543 \u2192 \u03b8+1"},"headline":"A181543 \u2192 \u03b8+1","chain":null}