{"id":"n-0624f1d2189c7352","revision":1,"first_index":0,"fingerprint":"f6-9a99cf3d0e7eaf56","name":"theta_plus_one_squared of Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 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"],["2","4"],["3","8"],["5","20","9"],["8","40","27"],["13","80","81","16"],["21","152","198","64"],["34","284","459","224","25"],["55","520","999","640","125"],["89","940","2097","1680","500","36"],["144","1680","4266","4096","1625","216"],["233","2976","8478","9504","4750","972","49"],["377","5232","16524","21184","12775","3528","343"],["610","9140","31698","45760","32375","11340","1715","64"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","6","11","34","75","190","435","1026","2339","5342","12027","26962","59963","132702"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0","0","0"],["100","85","9","-14","0"],["0","125","40","-8","0"],["0","0","75","25","1"],["0","0","0","15","2"],["0","0","0","0","1"]]},"terms":[{"coeff":{"coeffs":[["-200","-620","-668","-284","-44","-8"],["300","280","-268","-394","-142","-16"],["0","550","760","266","8","-2"],["0","0","300","375","125","14"],["0","0","0","55","54","10"],["0","0","0","0","3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-125","-525","-885","-771","-378","-108","-16"],["0","-375","-1250","-1645","-1078","-356","-48"],["0","0","-300","-815","-800","-337","-52"],["0","0","0","-75","-165","-114","-24"],["0","0","0","0","-5","-9","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161088,"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-a0717e73efaff775"]},{"id":161089,"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-a0717e73efaff775"]},{"id":161090,"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-a0717e73efaff775"]}],"region_exact":{},"incoming_edges":[{"id":"e-a0717e73efaff775","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-0624f1d2189c7352","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":"d474c8d9b1fd5bed5429d5f54c322f93a9cbd94c9393e8e76f6248dd2207b5a6","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-7d332fcb7bf9a0f7","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 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:  2,   4;\n  3:  3,   8;\n  4:  5,  20,   9;\n  5:  8,  40,  27;\n  6: 13,  80,  81,  16;\n  7: 21, 152, 198,  64;\n  8: 34, 284, 459, 224,  25;\n  9: 55, 520, 999, 640, 125;\n","recurrence_sentence":"The row polynomials satisfy (100*n+(125*n^2+85*n)*x+(75*n^3+40*n^2+9*n)*x^2+(15*n^4+25*n^3-8*n^2-14*n)*x^3+(n^5+2*n^4+n^3)*x^4)*P(n,x) = (300*n-200+(550*n^2+280*n-620)*x+(300*n^3+760*n^2-268*n-668)*x^2+(55*n^4+375*n^3+266*n^2-394*n-284)*x^3+(3*n^5+54*n^4+125*n^3+8*n^2-142*n-44)*x^4+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) - (125+(375*n+525)*x+(300*n^2+1250*n+885)*x^2+(75*n^3+815*n^2+1645*n+771)*x^3+(5*n^4+165*n^3+800*n^2+1078*n+378)*x^4+(9*n^4+114*n^3+337*n^2+356*n+108)*x^5+(4*n^4+24*n^3+52*n^2+48*n+16)*x^6)*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[((300*n - 200 + (550*n^2 + 280*n - 620)*x + (300*n^3 + 760*n^2 - 268*n - 668)*x^2 + (55*n^4 + 375*n^3 + 266*n^2 - 394*n - 284)*x^3 + (3*n^5 + 54*n^4 + 125*n^3 + 8*n^2 - 142*n - 44)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] - (125 + (375*n + 525)*x + (300*n^2 + 1250*n + 885)*x^2 + (75*n^3 + 815*n^2 + 1645*n + 771)*x^3 + (5*n^4 + 165*n^3 + 800*n^2 + 1078*n + 378)*x^4 + (9*n^4 + 114*n^3 + 337*n^2 + 356*n + 108)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(100*n + (125*n^2 + 85*n)*x + (75*n^3 + 40*n^2 + 9*n)*x^2 + (15*n^4 + 25*n^3 - 8*n^2 - 14*n)*x^3 + (n^5 + 2*n^4 + n^3)*x^4)]];\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] = ((300*n - 200 + (550*n^2 + 280*n - 620)*x + (300*n^3 + 760*n^2 - 268*n - 668)*x^2 + (55*n^4 + 375*n^3 + 266*n^2 - 394*n - 284)*x^3 + (3*n^5 + 54*n^4 + 125*n^3 + 8*n^2 - 142*n - 44)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] - (125 + (375*n + 525)*x + (300*n^2 + 1250*n + 885)*x^2 + (75*n^3 + 815*n^2 + 1645*n + 771)*x^3 + (5*n^4 + 165*n^3 + 800*n^2 + 1078*n + 378)*x^4 + (9*n^4 + 114*n^3 + 337*n^2 + 356*n + 108)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(100*n + (125*n^2 + 85*n)*x + (75*n^3 + 40*n^2 + 9*n)*x^2 + (15*n^4 + 25*n^3 - 8*n^2 - 14*n)*x^3 + (n^5 + 2*n^4 + n^3)*x^4)); 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-75ccd0453a9b6cec","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-2a29a8b9938e2bd7","e-a0717e73efaff775"],"text":"A011973 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A011973 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}