{"id":"n-7fb3d7e99783adbd","revision":0,"first_index":0,"fingerprint":"f6-2dbb8918a8b74f8e","name":"theta_plus_one of Triangle read by rows T(n, k) = binomial(2*n, k) * binomial(3*n - k, 2*n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","4"],["15","40","18"],["84","336","315","80"],["495","2640","3780","2016","350"],["3003","20020","38610","31680","11550","1512"],["18564","148512","360360","400400","225225","61776","6468"],["116280","1085280","3174444","4455360","3403400","1441440","315315","27456"],["735471","7845024","26860680","45581760","44089500","25395552","8576568","1555840","115830"],["4686825","56241900","220641300","439321344","514829700","376049520","172830840","48372480","7482618","486200"],["30045015","400600200","1771619850","4049416800","5577321750","4942365120","2883046320","1098303360","261891630","35271600","2032316"],["193536720","2838538560","13970931975","36054018000","57085528500","59526426960","42164552430","20397062400","6619239000","1372879200","163601438","8465184"],["1251677700","20026843200","108574099920","312239241600","558837279000","670604734800","559438179300","328812644160","135528918525","38244492000","7011490200","748843200","35154028"],["8122425444","140788707696","833617348200","2643543302400","5277907635000","7165890594720","6884875277280","4768744780800","2389029367725","856282927500","213499876590","35045861760","3389852700","145608400"],["52860229080","986724276160","6335491846320","21963038400576","48396118270500","73358326641600","79801963441200","63696805286400","37620925622280","16392560184000","5190977391600","1158317269200","172035024525","15201516960","601749000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","73","815","9281","106375","1221305","14018975","160756225","1840942727","21051913961","240394780367","2741353597633","31220883566215","355141947149401"],"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"],["2","0"],["-10","2"],["12","-8"],["0","8"]]},"terms":[{"coeff":{"coeffs":[["-6","-12","-6","0"],["45","82","35","0"],["-108","-165","-54","-1"],["81","63","-15","-1"],["0","54","48","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-6","-3","0"],["0","18","32","14","0"],["0","-27","-39","-13","-1"],["0","0","-18","-20","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184142,"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-8f4250389d4b375b"]},{"id":184143,"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-8f4250389d4b375b"]},{"id":184144,"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-8f4250389d4b375b"]}],"region_exact":{},"incoming_edges":[{"id":"e-8f4250389d4b375b","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-7fb3d7e99783adbd","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":"9b2dea89fe792972c136c22e29f4d5371959649ddf2bd46e6992b40a67a95ba5","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-1397bdceeefc7b3d","role":"sequence","name":"Triangle read by rows T(n, k) = binomial(2*n, k) * binomial(3*n - k, 2*n).","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:       3,        4;\n  2:      15,       40,        18;\n  3:      84,      336,       315,        80;\n  4:     495,     2640,      3780,      2016,       350;\n  5:    3003,    20020,     38610,     31680,     11550,      1512;\n  6:   18564,   148512,    360360,    400400,    225225,     61776,      6468;\n  7:  116280,  1085280,   3174444,   4455360,   3403400,   1441440,    315315,    27456;\n  8:  735471,  7845024,  26860680,  45581760,  44089500,  25395552,   8576568,  1555840,  115830;\n  9: 4686825, 56241900, 220641300, 439321344, 514829700, 376049520, 172830840, 48372480, 7482618, 486200;\n","recurrence_sentence":"The row polynomials satisfy (12*n^3-10*n^2+2*n+(8*n^4-8*n^3+2*n^2)*x)*P(n,x) = (81*n^3-108*n^2+45*n-6+(54*n^4+63*n^3-165*n^2+82*n-12)*x+(48*n^4-15*n^3-54*n^2+35*n-6)*x^2+(2*n^4-n^3-n^2)*x^3)*P(n-1,x) - ((27*n^2-18*n+3)*x+(18*n^3+39*n^2-32*n+6)*x^2+(20*n^3+13*n^2-14*n+3)*x^3+(2*n^3+n^2)*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[((81*n^3 - 108*n^2 + 45*n - 6 + (54*n^4 + 63*n^3 - 165*n^2 + 82*n - 12)*x + (48*n^4 - 15*n^3 - 54*n^2 + 35*n - 6)*x^2 + (2*n^4 - n^3 - n^2)*x^3)*P[n - 1, x] - ((27*n^2 - 18*n + 3)*x + (18*n^3 + 39*n^2 - 32*n + 6)*x^2 + (20*n^3 + 13*n^2 - 14*n + 3)*x^3 + (2*n^3 + n^2)*x^4)*D[P[n - 1, x], x])/(12*n^3 - 10*n^2 + 2*n + (8*n^4 - 8*n^3 + 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; for(n = 1, N, my(i = n + 1); v[i] = ((81*n^3 - 108*n^2 + 45*n - 6 + (54*n^4 + 63*n^3 - 165*n^2 + 82*n - 12)*x + (48*n^4 - 15*n^3 - 54*n^2 + 35*n - 6)*x^2 + (2*n^4 - n^3 - n^2)*x^3)*v[i - 1] - ((27*n^2 - 18*n + 3)*x + (18*n^3 + 39*n^2 - 32*n + 6)*x^2 + (20*n^3 + 13*n^2 - 14*n + 3)*x^3 + (2*n^3 + n^2)*x^4)*deriv(v[i - 1], x))/(12*n^3 - 10*n^2 + 2*n + (8*n^4 - 8*n^3 + 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":{"root":"n-1397bdceeefc7b3d","steps":["\u03b8+1"],"edges":["e-8f4250389d4b375b"],"text":"A357613 \u2192 \u03b8+1"},"headline":"A357613 \u2192 \u03b8+1","chain":null}