{"id":"n-7b7663b683c07e49","revision":0,"first_index":0,"fingerprint":"f6-880bd7c6b138382f","name":"theta_plus_one of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","4","3"],["1","12","30","24","5"],["1","28","141","272","235","84","7"],["1","60","528","1800","3030","2700","1232","240","9"],["1","124","1791","9568","25810","39672","36134","19136","5373","620","11"],["1","252","5778","45528","179835","414792","598444","553056","323703","113820","21186","1512","13"],["1","508","18129","203568","1118925","3603276","7422877","10216768","9543699","6005460","2461635","610704","78559","3556","15"],["1","1020","55956","876840","6496340","27969300","77599228","146744400","194774310","183430500","121941644","55938600","16890484","3068940","279780","8160","17"],["1","2044","170979","3689216","36072700","201820656","726689684","1798833664","3180618486","4101441800","3887422594","2698250496","1349566556","470914864","108218100","14756864","968881","18396","19"],["1","4092","519126","15286392","194430945","1386039600","6315269352","19818993216","44819041122","75064964040","94546469028","90077956848","64738614954","34683238128","13532720040","3696105600","661065213","68788764","3287798","40920","21"],["1","8188","1569717","62693584","1026809695","9195604452","52068434355","202555691904","569758585194","1197079890840","1917278844690","2368154169504","2265874998270","1675911847176","949597641990","405111383808","126451912005","27586813356","3901876841","313467920","10988019","90068","23"],["1","16380","4733784","255320520","5345722530","59518777500","413248588760","1960153606320","6721644781575","17288042935800","34169217113328","52711946043120","64043750655612","61497270383640","46594386972720","27660868697280","12696440142975","4410345614220","1121674740920","198395925000","22452034626","1404262860","36292344","196560","25"],["1","32764","14250567","1034691744","27548375670","378367976232","3188781511202","18212964453440","75000221316675","231920873903700","553555924278093","1039024796041152","1552505005595076","1859863956769392","1791351929532780","1385366394721536","855495519338871","417457573026660","158333800557425","45532411133600","9566344533606","1387349246184","126722528082","6208150464","118754725","425932","27"],["1","65532","42850074","4178472216","140922381335","2373572861784","24091386886196","164091153051168","802078262672769","2941200008854500","8341012895778838","18685546809056856","33555227215563807","48777081856000848","57725980650489240","55745236406858112","43879912512660363","28028320213585284","14407204092708902","5882400017709000","1871515946236461","451250670890712","79157414054644","9494291447136","704611906675","27160069404","385650666","917448","29"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","8","72","768","9600","138240","2257920","41287680","836075520","18579456000","449622835200","11771943321600","331576403558400","9998303861145600","321374052679680000"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","4","3"]],"recurrence":{"denominator":{"coeffs":[["1","-2"],["0","2"]]},"terms":[{"coeff":{"coeffs":[["0","0","-6","12","-6","0"],["0","0","2","-20","34","-12"],["0","0","0","4","-20","20"],["0","0","0","0","0","-8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2","2","2","-2","0"],["0","0","0","-6","4","6","-4"],["0","0","0","0","-4","0","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2","-9","6"],["0","2","12","-14"],["0","0","0","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-2","-1","2"],["0","0","2","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185411,"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-60ceafad0998b436"]},{"id":185412,"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-60ceafad0998b436"]},{"id":185413,"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-60ceafad0998b436"]}],"region_exact":{},"incoming_edges":[{"id":"e-60ceafad0998b436","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-7b7663b683c07e49","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":"55e76063f7c95ddb6e850c8aaa1e6689077bd380a6e9de3722e5c811b0620e91","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-b3367e055aead099","role":"sequence","name":"Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), 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,    4,      3;\n  2: 1,   12,     30,      24,        5;\n  3: 1,   28,    141,     272,      235,        84,         7;\n  4: 1,   60,    528,    1800,     3030,      2700,      1232,        240,          9;\n  5: 1,  124,   1791,    9568,    25810,     39672,     36134,      19136,       5373,        620,         11;\n  6: 1,  252,   5778,   45528,   179835,    414792,    598444,     553056,     323703,     113820,      21186,       1512,         13;\n  7: 1,  508,  18129,  203568,  1118925,   3603276,   7422877,   10216768,    9543699,    6005460,    2461635,     610704,      78559,      3556,        15;\n  8: 1, 1020,  55956,  876840,  6496340,  27969300,  77599228,  146744400,  194774310,  183430500,  121941644,   55938600,   16890484,   3068940,    279780,     8160,     17;\n  9: 1, 2044, 170979, 3689216, 36072700, 201820656, 726689684, 1798833664, 3180618486, 4101441800, 3887422594, 2698250496, 1349566556, 470914864, 108218100, 14756864, 968881, 18396, 19;\n","recurrence_sentence":"The row polynomials satisfy (1+(2*n-2)*x)*P(n,x) = ((2*n-6)*x^2+(4*n^2-20*n+12)*x^3-(20*n^2-34*n+6)*x^4-(8*n^3-20*n^2+12*n)*x^5)*P(n-2,x) - (2*x^2+(6*n-2)*x^3+(4*n^2-4*n-2)*x^4-(6*n-2)*x^5-(4*n^2-4*n)*x^6)*d/dx P(n-2,x) + (1+(2*n+2)*x+(12*n-9)*x^2+(8*n^2-14*n+6)*x^3)*P(n-1,x) + (x+(2*n-2)*x^2-x^3-(2*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(((2*n - 6)*x^2 + (4*n^2 - 20*n + 12)*x^3 - (20*n^2 - 34*n + 6)*x^4 - (8*n^3 - 20*n^2 + 12*n)*x^5)*P[n - 2, x] - (2*x^2 + (6*n - 2)*x^3 + (4*n^2 - 4*n - 2)*x^4 - (6*n - 2)*x^5 - (4*n^2 - 4*n)*x^6)*D[P[n - 2, x], x] + (1 + (2*n + 2)*x + (12*n - 9)*x^2 + (8*n^2 - 14*n + 6)*x^3)*P[n - 1, x] + (x + (2*n - 2)*x^2 - x^3 - (2*n - 2)*x^4)*D[P[n - 1, x], x])/(1 + (2*n - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 6)*x^2 + (4*n^2 - 20*n + 12)*x^3 - (20*n^2 - 34*n + 6)*x^4 - (8*n^3 - 20*n^2 + 12*n)*x^5)*v[i - 2] - (2*x^2 + (6*n - 2)*x^3 + (4*n^2 - 4*n - 2)*x^4 - (6*n - 2)*x^5 - (4*n^2 - 4*n)*x^6)*deriv(v[i - 2], x) + (1 + (2*n + 2)*x + (12*n - 9)*x^2 + (8*n^2 - 14*n + 6)*x^3)*v[i - 1] + (x + (2*n - 2)*x^2 - x^3 - (2*n - 2)*x^4)*deriv(v[i - 1], x))/(1 + (2*n - 2)*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":null,"headline":"theta_plus_one of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.","chain":null}