{"id":"n-c823c2082410ea42","revision":0,"first_index":0,"fingerprint":"f6-0631f52eb5bb8a38","name":"theta_plus_one of basis_map_A013609 of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["3","8"],["5","16"],["11","56","48"],["21","128","144"],["43","328","576","256"],["85","752","1632","1024"],["171","1752","4752","4608","1280"],["341","3936","12528","15360","6400"],["683","8808","32544","49920","32000","6144"],["1365","19408","81216","147456","121600","36864"],["2731","42488","199152","420864","435200","202752","28672"],["5461","92224","478032","1148928","1415680","860160","200704"],["10923","199048","1131264","3052800","4390400","3354624","1204224","131072"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","11","21","115","293","1203","3493","12563","38565","130099","407909","1331859","4201189","13474355"],"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"],["36","52","0"],["0","36","16"],["0","0","16"]]},"terms":[{"coeff":{"coeffs":[["-144","-640","-832","-256"],["180","476","192","-128"],["0","252","592","256"],["0","0","80","128"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-81","-540","-1316","-1376","-512"],["0","-162","-828","-1392","-768"],["0","0","-72","-272","-256"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173804,"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-a9dcf31da9658708"]},{"id":173805,"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-a9dcf31da9658708"]},{"id":173806,"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-a9dcf31da9658708"]}],"region_exact":{},"incoming_edges":[{"id":"e-a9dcf31da9658708","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-c823c2082410ea42","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":"59db368e2a12658be744ca352c9fbb131fd7ef8f29414163882b8d7d8cecd77e","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-9e53622e0c440ce5","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).","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;\n  2:   3,    8;\n  3:   5,   16;\n  4:  11,   56,    48;\n  5:  21,  128,   144;\n  6:  43,  328,   576,   256;\n  7:  85,  752,  1632,  1024;\n  8: 171, 1752,  4752,  4608, 1280;\n  9: 341, 3936, 12528, 15360, 6400;\n","recurrence_sentence":"The row polynomials satisfy (36*n+(36*n^2+52*n)*x+(16*n^3+16*n^2)*x^2)*P(n,x) = (180*n-144+(252*n^2+476*n-640)*x+(80*n^3+592*n^2+192*n-832)*x^2+(128*n^3+256*n^2-128*n-256)*x^3)*P(n-1,x) - (81+(162*n+540)*x+(72*n^2+828*n+1316)*x^2+(272*n^2+1392*n+1376)*x^3+(256*n^2+768*n+512)*x^4)*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[((180*n - 144 + (252*n^2 + 476*n - 640)*x + (80*n^3 + 592*n^2 + 192*n - 832)*x^2 + (128*n^3 + 256*n^2 - 128*n - 256)*x^3)*P[n - 1, x] - (81 + (162*n + 540)*x + (72*n^2 + 828*n + 1316)*x^2 + (272*n^2 + 1392*n + 1376)*x^3 + (256*n^2 + 768*n + 512)*x^4)*D[P[n - 1, x], x])/(36*n + (36*n^2 + 52*n)*x + (16*n^3 + 16*n^2)*x^2)]];\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] = ((180*n - 144 + (252*n^2 + 476*n - 640)*x + (80*n^3 + 592*n^2 + 192*n - 832)*x^2 + (128*n^3 + 256*n^2 - 128*n - 256)*x^3)*v[i - 1] - (81 + (162*n + 540)*x + (72*n^2 + 828*n + 1316)*x^2 + (272*n^2 + 1392*n + 1376)*x^3 + (256*n^2 + 768*n + 512)*x^4)*deriv(v[i - 1], x))/(36*n + (36*n^2 + 52*n)*x + (16*n^3 + 16*n^2)*x^2)); 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-e3d901814302c059","steps":["basis A013609","\u03b8+1"],"edges":["e-2ee1895757bd17cf","e-a9dcf31da9658708"],"text":"A128099 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A128099 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}