{"id":"n-62041b6e495f849d","revision":1,"first_index":1,"fingerprint":"f6-7ce493974858d62c","name":"theta_plus_one_squared of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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"],["0","4"],["0","8","9"],["0","8","36","16"],["0","8","72","96","25"],["0","8","108","288","200","36"],["0","8","144","608","800","360","49"],["0","8","180","1056","2200","1800","588","64"],["0","8","216","1632","4800","6120","3528","896","81"],["0","8","252","2336","9000","16200","14308","6272","1296","100"],["0","8","288","3168","15200","36072","44688","29568","10368","1800","121"],["0","8","324","4128","23800","70920","115836","106624","55728","16200","2420","144"],["0","8","360","5216","35200","127080","261464","316288","228096","97800","24200","3168","169"],["0","8","396","6432","49800","212040","530964","809088","763344","448200","162140","34848","4056","196"],["0","8","432","7776","68000","334440","992544","1844096","2187648","1672200","822800","256608","48672","5096","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","17","60","201","640","1969","5896","17281","49772","141281","396132","1099049","3021512","8240545"],"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"],["0","4"],["0","8","9"],["0","8","36","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","9"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","4"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","-4","18"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","4","8"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-1","0","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161320,"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-75459cd887e4a44f"]},{"id":161321,"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-75459cd887e4a44f"]},{"id":161322,"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-75459cd887e4a44f"]},{"id":161323,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-75459cd887e4a44f"]}],"region_exact":{},"incoming_edges":[{"id":"e-75459cd887e4a44f","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-62041b6e495f849d","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":"f1fe3338a08affc51fdd0ee5532ca7ae060715b9b7f5ebdcbdcf9d83384e4566","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b0fc620949fb5990","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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: 0, 4;\n   3: 0, 8,   9;\n   4: 0, 8,  36,   16;\n   5: 0, 8,  72,   96,   25;\n   6: 0, 8, 108,  288,  200,    36;\n   7: 0, 8, 144,  608,  800,   360,    49;\n   8: 0, 8, 180, 1056, 2200,  1800,   588,   64;\n   9: 0, 8, 216, 1632, 4800,  6120,  3528,  896,   81;\n  10: 0, 8, 252, 2336, 9000, 16200, 14308, 6272, 1296, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 9*x^2*P(n-4,x) + 4*x^3*d/dx P(n-4,x) - (4*x-18*x^2)*P(n-3,x) + (4*x^2+8*x^3)*d/dx P(n-3,x) - (1-9*x^2)*P(n-2,x) + 4*x^3*d/dx P(n-2,x) + (2+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 4*x, P(3,x) = 8*x + 9*x^2, P(4,x) = 8*x + 36*x^2 + 16*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4*x;\nP[3, x_] := 8*x + 9*x^2;\nP[4, x_] := 8*x + 36*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[9*x^2*P[n - 4, x] + 4*x^3*D[P[n - 4, x], x] - (4*x - 18*x^2)*P[n - 3, x] + (4*x^2 + 8*x^3)*D[P[n - 3, x], x] - (1 - 9*x^2)*P[n - 2, x] + 4*x^3*D[P[n - 2, x], x] + (2 + 4*x)*P[n - 1, 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] = 4*x; v[3] = 8*x + 9*x^2; v[4] = 8*x + 36*x^2 + 16*x^3; for(n = 5, N, my(i = n); v[i] = 9*x^2*v[i - 4] + 4*x^3*deriv(v[i - 4], x) - (4*x - 18*x^2)*v[i - 3] + (4*x^2 + 8*x^3)*deriv(v[i - 3], x) - (1 - 9*x^2)*v[i - 2] + 4*x^3*deriv(v[i - 2], x) + (2 + 4*x)*v[i - 1] - 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-b0fc620949fb5990","steps":["(\u03b8+1)\u00b2"],"edges":["e-75459cd887e4a44f"],"text":"A122542 \u2192 (\u03b8+1)\u00b2"},"headline":"A122542 \u2192 (\u03b8+1)\u00b2","chain":null}