{"id":"n-bd89bcbd060a3e48","revision":0,"first_index":1,"fingerprint":"f6-c9871221535ca904","name":"theta_plus_one of Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","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","2"],["0","4","9"],["0","6","36","40"],["0","8","90","240","175"],["0","10","180","840","1400","756"],["0","12","315","2240","6300","7560","3234"],["0","14","504","5040","21000","41580","38808","13728"],["0","16","756","10080","57750","166320","252252","192192","57915"],["0","18","1080","18480","138600","540540","1177176","1441440","926640","243100"],["0","20","1485","31680","300300","1513512","4414410","7687680","7876440","4375800","1016158"],["0","22","1980","51480","600600","3783780","14126112","32672640","47258640","41570100","20323160","4232592"],["0","24","2574","80080","1126125","8648640","40023984","117621504","224478540","277134000","213393180","93117024","17577014"],["0","26","3276","120120","2002000","18378360","102918816","372468096","897914160","1454953500","1564883320","1070845776","421848336","72804200"],["0","28","4095","174720","3403400","36756720","244432188","1064194560","3142699560","6401795400","8998079090","8566766208","5273104200","1892909200","300874500"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","13","82","513","3186","19661","120674","737281","4487074","27217485","164621106","993202689","5979139986","35925193869"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["0","2"],["0","4","9"],["0","6","36","40"]],"recurrence":{"denominator":{"coeffs":[["126"],["-253"],["167"],["-44"],["4"]]},"terms":[{"coeff":{"coeffs":[["-300"],["455"],["-239"],["52"],["-4"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1104","2160"],["-1684","-3324"],["896","1784"],["-200","-400"],["16","32"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-1458","-3888","-3780"],["2268","6048","5952"],["-1242","-3312","-3296"],["288","768","768"],["-24","-64","-64"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["792","1512"],["-1292","-2532"],["752","1496"],["-184","-368"],["16","32"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177325,"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-4de0d606ad5c4b3e"]},{"id":177326,"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-4de0d606ad5c4b3e"]},{"id":177327,"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-4de0d606ad5c4b3e"]}],"region_exact":{},"incoming_edges":[{"id":"e-4de0d606ad5c4b3e","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-bd89bcbd060a3e48","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":"28b125bb145bcb2a3da0beb910d96cb0fb354d2a1d6dc9e798e300aaafd20f5f","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-b0d6ec97a8d172b8","role":"sequence","name":"Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","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,  2;\n   3: 0,  4,    9;\n   4: 0,  6,   36,    40;\n   5: 0,  8,   90,   240,    175;\n   6: 0, 10,  180,   840,   1400,    756;\n   7: 0, 12,  315,  2240,   6300,   7560,    3234;\n   8: 0, 14,  504,  5040,  21000,  41580,   38808,   13728;\n   9: 0, 16,  756, 10080,  57750, 166320,  252252,  192192,  57915;\n  10: 0, 18, 1080, 18480, 138600, 540540, 1177176, 1441440, 926640, 243100;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4-44*n^3+167*n^2-253*n+126)*P(n,x) = -(4*n^4-52*n^3+239*n^2-455*n+300)*P(n-4,x) + (16*n^4-200*n^3+896*n^2-1684*n+1104+(32*n^4-400*n^3+1784*n^2-3324*n+2160)*x)*P(n-3,x) - (24*n^4-288*n^3+1242*n^2-2268*n+1458+(64*n^4-768*n^3+3312*n^2-6048*n+3888)*x+(64*n^4-768*n^3+3296*n^2-5952*n+3780)*x^2)*P(n-2,x) + (16*n^4-184*n^3+752*n^2-1292*n+792+(32*n^4-368*n^3+1496*n^2-2532*n+1512)*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2*x, P(3,x) = 4*x + 9*x^2, P(4,x) = 6*x + 36*x^2 + 40*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2*x;\nP[3, x_] := 4*x + 9*x^2;\nP[4, x_] := 6*x + 36*x^2 + 40*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*P[n - 4, x] + (16*n^4 - 200*n^3 + 896*n^2 - 1684*n + 1104 + (32*n^4 - 400*n^3 + 1784*n^2 - 3324*n + 2160)*x)*P[n - 3, x] - (24*n^4 - 288*n^3 + 1242*n^2 - 2268*n + 1458 + (64*n^4 - 768*n^3 + 3312*n^2 - 6048*n + 3888)*x + (64*n^4 - 768*n^3 + 3296*n^2 - 5952*n + 3780)*x^2)*P[n - 2, x] + (16*n^4 - 184*n^3 + 752*n^2 - 1292*n + 792 + (32*n^4 - 368*n^3 + 1496*n^2 - 2532*n + 1512)*x)*P[n - 1, x])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; v[3] = 4*x + 9*x^2; v[4] = 6*x + 36*x^2 + 40*x^3; for(n = 5, N, my(i = n); v[i] = (-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*v[i - 4] + (16*n^4 - 200*n^3 + 896*n^2 - 1684*n + 1104 + (32*n^4 - 400*n^3 + 1784*n^2 - 3324*n + 2160)*x)*v[i - 3] - (24*n^4 - 288*n^3 + 1242*n^2 - 2268*n + 1458 + (64*n^4 - 768*n^3 + 3312*n^2 - 6048*n + 3888)*x + (64*n^4 - 768*n^3 + 3296*n^2 - 5952*n + 3780)*x^2)*v[i - 2] + (16*n^4 - 184*n^3 + 752*n^2 - 1292*n + 792 + (32*n^4 - 368*n^3 + 1496*n^2 - 2532*n + 1512)*x)*v[i - 1])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)); 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-b0d6ec97a8d172b8","steps":["\u03b8+1"],"edges":["e-4de0d606ad5c4b3e"],"text":"A253283 \u2192 \u03b8+1"},"headline":"A253283 \u2192 \u03b8+1","chain":null}