{"id":"n-9f302d2b05fb08ea","revision":0,"first_index":0,"fingerprint":"f6-a684a258783da613","name":"theta_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.","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"],["1","2"],["1","40","3"],["1","438","657","4"],["1","4436","24966","8872","5"],["1","44434","660666","880888","111085","6"],["1","444432","15018741","49329728","25031235","1333296","7"],["1","4444430","316021887","2122156940","2652696175","632043774","15555505","8"],["1","44444428","6371081328","78803070056","196299516950","118204605084","14865856432","177777712","9"],["1","444444426","125317210320","2673698592288","11794258655370","14153110386444","4678972536504","334179227520","1999999917","10"],["1","4444444424","2429693660727","85557295864704","620204348468490","1302086155552848","868286087855886","171114591729408","7289080982181","22222222120","11"],["1","44444444422","46710846217965","2632094467189140","29792144562510450","100829450154351348","117634358513409906","47667431300016720","5922212551175565","155702820726550","244444444321","12"],["1","444444444420","893572744804938","78805697601126760","1342487968943895675","6926191409500964568","12939779436475089660","9234921879334619424","2416478344099012215","197014244002816900","3276433397618106","2666666666520","13"],["1","4444444444418","17044548817956822","2314979692534548424","57749638855039632875","436206750916529631258","1228843494248868619044","1404392564855849850336","654310126374794446887","115499277710079265750","6366194154470008166","68178195271827288","28888888888717","14"],["1","44444444444416","324573094207841961","67092037900028265472","2400065577662271299605","25748075005496262912768","104736667136745547642893","179762248301548780843008","134661429175815704112291","42913458342493771521280","5280144270856996859131","201276113700084796416","1406483408233981831","311111111110912","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","44","1100","38280","1697080","91157440","5742918720","414766352000","33761981052800","3056993920780800","304680394104486400","33139852097211059200","3905768282885589120000","495772187374277445632000"],"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","2"]],"recurrence":{"denominator":{"coeffs":[["-8"],["9"]]},"terms":[{"coeff":{"coeffs":[["8","-272","-136"],["72","-2304","-999"],["0","1296","1944"],["0","0","-729"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","72","0","-72"],["0","648","-81","-567"],["0","0","-729","729"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-16","128"],["-63","-288"],["0","162"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-72","72"],["0","81","-81"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191782,"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-2fe8b900fc9e4205"]},{"id":191783,"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-2fe8b900fc9e4205"]},{"id":191784,"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-2fe8b900fc9e4205"]},{"id":191785,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-2fe8b900fc9e4205"]}],"region_exact":{},"incoming_edges":[{"id":"e-2fe8b900fc9e4205","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-9f302d2b05fb08ea","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":"9c0681e7e7bc5f327e4bfa74e755c52b768ac68ac0afafb4efe8411c5b22dd6e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-dcd2d11144e6ea21","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.","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,         2;\n  2: 1,        40,            3;\n  3: 1,       438,          657,             4;\n  4: 1,      4436,        24966,          8872,              5;\n  5: 1,     44434,       660666,        880888,         111085,              6;\n  6: 1,    444432,     15018741,      49329728,       25031235,        1333296,             7;\n  7: 1,   4444430,    316021887,    2122156940,     2652696175,      632043774,      15555505,            8;\n  8: 1,  44444428,   6371081328,   78803070056,   196299516950,   118204605084,   14865856432,    177777712,          9;\n  9: 1, 444444426, 125317210320, 2673698592288, 11794258655370, 14153110386444, 4678972536504, 334179227520, 1999999917, 10;\n","recurrence_sentence":"The row polynomials satisfy (9*n-8)*P(n,x) = (72*n+8+(1296*n^2-2304*n-272)*x-(729*n^3-1944*n^2+999*n+136)*x^2)*P(n-2,x) + ((648*n+72)*x-(729*n^2+81*n)*x^2+(729*n^2-567*n-72)*x^3)*d/dx P(n-2,x) - (63*n+16-(162*n^2-288*n+128)*x)*P(n-1,x) + ((81*n-72)*x-(81*n-72)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((72*n + 8 + (1296*n^2 - 2304*n - 272)*x - (729*n^3 - 1944*n^2 + 999*n + 136)*x^2)*P[n - 2, x] + ((648*n + 72)*x - (729*n^2 + 81*n)*x^2 + (729*n^2 - 567*n - 72)*x^3)*D[P[n - 2, x], x] - (63*n + 16 - (162*n^2 - 288*n + 128)*x)*P[n - 1, x] + ((81*n - 72)*x - (81*n - 72)*x^2)*D[P[n - 1, x], x])/(9*n - 8)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((72*n + 8 + (1296*n^2 - 2304*n - 272)*x - (729*n^3 - 1944*n^2 + 999*n + 136)*x^2)*v[i - 2] + ((648*n + 72)*x - (729*n^2 + 81*n)*x^2 + (729*n^2 - 567*n - 72)*x^3)*deriv(v[i - 2], x) - (63*n + 16 - (162*n^2 - 288*n + 128)*x)*v[i - 1] + ((81*n - 72)*x - (81*n - 72)*x^2)*deriv(v[i - 1], x))/(9*n - 8)); 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-dcd2d11144e6ea21","steps":["\u03b8+1"],"edges":["e-2fe8b900fc9e4205"],"text":"A257608 \u2192 \u03b8+1"},"headline":"A257608 \u2192 \u03b8+1","chain":null}