{"id":"n-05b91207a1ba3acf","revision":1,"first_index":1,"fingerprint":"f6-509399d8a74ba170","name":"Triangular array: T(n,k) = sqrt(C(n-1,k-1)*C(n-1,k)*C(n,k+1)* C(n+1,k+1)*C(n+1,k)*C(n,k-1)), where C(n,k) = binomial(n,k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":false,"rows":[["30","120","30"],["60","600","600","60"],["105","2100","5250","2100","105"],["168","5880","29400","29400","5880","168"],["252","14112","123480","246960","123480","14112","252"],["360","30240","423360","1481760","1481760","423360","30240","360"],["495","59400","1247400","6985440","12224520","6985440","1247400","59400","495"],["660","108900","3267000","27442800","76839840","76839840","27442800","3267000","108900","660"],["858","188760","7786350","93436200","392432040","627891264","392432040","93436200","7786350","188760","858"],["1092","312312","17177160","283423140","1700538840","4081293216","4081293216","1700538840","283423140","17177160","312312","1092"],["1365","496860","35525490","781560780","6447876435","22107004920","33160507380","22107004920","6447876435","781560780","35525490","496860","1365"],["1680","764400","69560400","1989427440","21883701840","103166022960","221070049200","221070049200","103166022960","21883701840","1989427440","69560400","764400","1680"],["2040","1142400","129948000","4730107200","67640532960","425169064320","1252730278800","1789614684000","1252730278800","425169064320","67640532960","4730107200","129948000","1142400","2040"],["2448","1664640","233049600","10603756800","192988373760","1576990711296","6195320651520","12169379851200","12169379851200","6195320651520","1576990711296","192988373760","10603756800","233049600","1664640","2448"],["2907","2372112","403259040","22582506240","513752016960","5343020976384","27287571415104","71467448944320","98267742298440","71467448944320","27287571415104","5343020976384","513752016960","22582506240","403259040","2372112","2907"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["180","1320","9660","70896","522648","3871440","28809990","215318400","1615579680","12165491520","91905439080","696359055840","5290416835440","40291036122528","307537305284574"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A197208","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["30","120","30"],["60","600","600","60"]],"recurrence":{"denominator":{"coeffs":[["1"],["6/5"],["1/5"]]},"terms":[{"coeff":{"coeffs":[["4","4"],["7/5","28/5"],["-1/5","8/5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","21/5","-21/5"],["0","9/5","-9/5"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["6/5","12/5","6/5"],["8/5","16/5","8/5"],["2/5","4/5","2/5"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160394,"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-aaeb3397827b8bbd"]},{"id":160395,"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-aaeb3397827b8bbd"]},{"id":160396,"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-aaeb3397827b8bbd"]}],"region_exact":{},"incoming_edges":[{"id":"e-aaeb3397827b8bbd","operation":{"index_shift":2,"operation":"preserver_image","operator":"{\"shape\":\"hadamard_multiplier\",\"factor\":{\"terms\":[{\"kind\":\"binomial\",\"index_offset\":2,\"coefficient_offset\":1}]}}"},"operation_detail":"hadamard_multiplier, shift 2","conclusion_id":"n-05b91207a1ba3acf","premise_count":1,"theorem":"Polya--Schur characterization of multiplier sequences (G. Polya and I. Schur, J. Reine Angew. Math. 144 (1914) 89--113; stated as Theorem 40 of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601): a real sequence (lambda_k) is a multiplier sequence (x^k -> lambda_k x^k maps real-rooted polynomials to real-rooted polynomials or zero) iff sum_k binom(m,k) lambda_k x^k is real-rooted with zeros of one sign for every m, iff sum_k lambda_k x^k / k! (or its reflection) is C x^m e^(a x) prod (1 + alpha_j x) with a, alpha_j >= 0; a product of multiplier sequences is one (composition), and a nonzero factor lambda(n) of the row changes nothing","lean_route":null,"lean_declarations":[],"certificate_hash":"ae7baffb4fb7f45b3e62a1927fda890eb1941f01579fafb7f0af89ec9600f404","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-c963abfc0266d66c","role":"sequence","name":"Triangle of Dyck paths counted by number of long interior inclines.","uses":{"claim":"real_rooted","from_index":3,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 3, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-3f6abdcdad6c3c63","kind":"fingerprint_related","node_a_id":"n-047228a38014e6d4","node_b_id":"n-05b91207a1ba3acf","fingerprint":"f6-509399d8a74ba170","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-509399d8a74ba170"},"other_id":"n-047228a38014e6d4","other":{"id":"n-047228a38014e6d4","name":"Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","first_index":0,"fingerprint":"f6-509399d8a74ba170","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","4","1"]],"card_rows":[["1"],["1","1"],["1","4","1"],["1","10","10","1"]],"row_sums_preview":["1","2","6","22","92","422","2074","10754"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A056939"],"derivation":{"root":"n-c963abfc0266d66c","steps":["\u2299 C(n+1,k+1) (shift +1)"],"edges":["e-548c39a476f786f5"],"text":"A108838 \u2192 \u2299 C(n+1,k+1) (shift +1)"},"headline":"A056939: Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","chain":"A108838 \u2192 \u2299 C(n+1,k+1) (shift +1)"}},{"id":"l-dccbf91ad0750a94","kind":"oeis_match","node_a_id":"n-05b91207a1ba3acf","node_b_id":null,"fingerprint":null,"oeis_id":"A197209","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":2,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A197209","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A197209","relation":"match_row_sums"},{"oeis_id":"A197208","relation":"tail"}],"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:   30,    120,       30;\n   2:   60,    600,      600,        60;\n   3:  105,   2100,     5250,      2100,        105;\n   4:  168,   5880,    29400,     29400,       5880,        168;\n   5:  252,  14112,   123480,    246960,     123480,      14112,        252;\n   6:  360,  30240,   423360,   1481760,    1481760,     423360,      30240,        360;\n   7:  495,  59400,  1247400,   6985440,   12224520,    6985440,    1247400,      59400,       495;\n   8:  660, 108900,  3267000,  27442800,   76839840,   76839840,   27442800,    3267000,    108900,      660;\n   9:  858, 188760,  7786350,  93436200,  392432040,  627891264,  392432040,   93436200,   7786350,   188760,    858;\n  10: 1092, 312312, 17177160, 283423140, 1700538840, 4081293216, 4081293216, 1700538840, 283423140, 17177160, 312312, 1092;\n","recurrence_sentence":"The row polynomials satisfy ((1/5)*n^2+(6/5)*n+1)*P(n,x) = -((1/5)*n^2-(7/5)*n-4-((8/5)*n^2+(28/5)*n+4)*x)*P(n-1,x) + (((9/5)*n+21/5)*x-((9/5)*n+21/5)*x^2)*d/dx P(n-1,x) + ((2/5)*n^2+(8/5)*n+6/5+((4/5)*n^2+(16/5)*n+12/5)*x+((2/5)*n^2+(8/5)*n+6/5)*x^2)*P(n-2,x) for n >= 3, with P(1,x) = 30 + 120*x + 30*x^2, P(2,x) = 60 + 600*x + 600*x^2 + 60*x^3.","mathematica":"P[1, x_] := 30 + 120*x + 30*x^2;\nP[2, x_] := 60 + 600*x + 600*x^2 + 60*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-((1/5)*n^2 - (7/5)*n - 4 - ((8/5)*n^2 + (28/5)*n + 4)*x)*P[n - 1, x] + (((9/5)*n + 21/5)*x - ((9/5)*n + 21/5)*x^2)*D[P[n - 1, x], x] + ((2/5)*n^2 + (8/5)*n + 6/5 + ((4/5)*n^2 + (16/5)*n + 12/5)*x + ((2/5)*n^2 + (8/5)*n + 6/5)*x^2)*P[n - 2, x])/((1/5)*n^2 + (6/5)*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 30 + 120*x + 30*x^2; v[2] = 60 + 600*x + 600*x^2 + 60*x^3; for(n = 3, N, my(i = n); v[i] = (-((1/5)*n^2 - (7/5)*n - 4 - ((8/5)*n^2 + (28/5)*n + 4)*x)*v[i - 1] + (((9/5)*n + 21/5)*x - ((9/5)*n + 21/5)*x^2)*deriv(v[i - 1], x) + ((2/5)*n^2 + (8/5)*n + 6/5 + ((4/5)*n^2 + (16/5)*n + 12/5)*x + ((2/5)*n^2 + (8/5)*n + 6/5)*x^2)*v[i - 2])/((1/5)*n^2 + (6/5)*n + 1)); 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-c963abfc0266d66c","steps":["\u2299 C(n+2,k+1) (shift +2)"],"edges":["e-aaeb3397827b8bbd"],"text":"A108838 \u2192 \u2299 C(n+2,k+1) (shift +2)"},"headline":"A108838 \u2192 \u2299 C(n+2,k+1) (shift +2)","chain":null}