{"id":"n-bdd4abd52f61e954","revision":0,"first_index":0,"fingerprint":"f6-02dd92f09da9263e","name":"basis_map_A013610 of basis_map_A013611 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.","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":true,"rows":[["1"],["6","12"],["54","180","144"],["648","2808","3888","1728"],["9720","49896","92016","72576","20736"],["174960","1014768","2255040","2410560","1244160","248832"],["3674160","23409648","59533056","77682240","55054080","20155392","2985984"],["88179840","605921472","1709709120","2578770432","2253484800","1144378368","313528320","35831808"],["2380855680","17418037824","53433203904","90143311104","91789334784","57940033536","22197805056","4729798656","429981696"],["71425670400","551111402880","1812012571008","3345497779968","3835399776768","2839673023488","1361214554112","408267620352","69657034752","5159780352"],["2357047123200","19043784339840","66409751677824","132145577591040","166714165992960","139734007096320","78996156567552","29807406120960","7197893591040","1006157168640","61917364224"],["84853696435200","713860801712640","2619276472479744","5554157813411328","7587456906839040","7030994247383040","4520669721587712","2021020499165184","616813042728960","122596381163520","14302911135744","743008370688"],["3309294160972800","28858815624015360","110718112047261696","248043472392798720","362560713127658496","365258258530007040","260678050110517248","133067836126494720","48307954656411648","12183015378124800","2028970108256256","200612260085760","8916100448256"],["138990354760857600","1251781786140318720","4996466493473175552","11746443185064686592","18204071620075241472","19691575415792197632","15331577207001808896","8716985718638985216","3625748129087225856","1091382101758181376","231412929084260352","32773356222676992","2781823339855872","106993205379072"],["6254565964238592000","57998064633444633600","239862373639976724480","588547541249589003264","960140541124162105344","1104569753151551791104","926219879304587771904","576243283822776041472","267762494432592986112","92621172128164872192","23510167029889892352","4251756179031588864","518462324965638144","38196574320328704","1283918464548864"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","378","9072","244944","7348320","242494560","8729804160","340462362240","14299419214080","643473864633600","30886745502412800","1575224020623052800","85062097113644851200","4848539535477756518400"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","12"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203493,"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-58fd3bd2ee248a68","e-6bb7ade965ee86e7"]},{"id":203494,"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-58fd3bd2ee248a68","e-6bb7ade965ee86e7"]},{"id":203495,"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-58fd3bd2ee248a68","e-6bb7ade965ee86e7"]},{"id":203496,"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-58fd3bd2ee248a68","e-6bb7ade965ee86e7"]}],"region_exact":{},"incoming_edges":[{"id":"e-58fd3bd2ee248a68","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-bdd4abd52f61e954","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"28dd25ca13b76b5a31588a698d0631bb0927042d0c7a8bb5f39edb5dc5a2129a","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-7c95902cf1293962","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.","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]"}]},{"id":"e-6bb7ade965ee86e7","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-bdd4abd52f61e954","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"8e5ae241a32b0cab4e38ccfb2cfb2fffd39f53aba5ce5a37e2dfba37255dee29","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-5b94261d6f603245","role":"sequence","name":"basis_map_A013610 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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":[{"id":"l-24f3329de19bba88","kind":"fingerprint_related","node_a_id":"n-5f097fd6f4e6ec91","node_b_id":"n-bdd4abd52f61e954","fingerprint":"f6-02dd92f09da9263e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02dd92f09da9263e"},"other_id":"n-5f097fd6f4e6ec91","other":{"id":"n-5f097fd6f4e6ec91","name":"reversal of basis_map_A013609 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","first_index":0,"fingerprint":"f6-02dd92f09da9263e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["16","20","6"]],"card_rows":[["1"],["4","2"],["16","20","6"],["64","144","104","24"]],"row_sums_preview":["1","6","42","336","3024","30240","332640","3991680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-60c7e36cbe887b7e","steps":["basis A013609","reverse"],"edges":["e-7070cc8d62757598","e-79eec1a2b6ee56f0"],"text":"A137320 \u2192 basis A013609 \u2192 reverse"},"headline":"A137320 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-a366eb8a9ae2d5d1","kind":"fingerprint_related","node_a_id":"n-52d9fa43425cb9e1","node_b_id":"n-bdd4abd52f61e954","fingerprint":"f6-02dd92f09da9263e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02dd92f09da9263e"},"other_id":"n-52d9fa43425cb9e1","other":{"id":"n-52d9fa43425cb9e1","name":"basis_map_A013609 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","first_index":0,"fingerprint":"f6-02dd92f09da9263e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["6","20","16"]],"card_rows":[["1"],["2","4"],["6","20","16"],["24","104","144","64"]],"row_sums_preview":["1","6","42","336","3024","30240","332640","3991680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-60c7e36cbe887b7e","steps":["basis A013609"],"edges":["e-7070cc8d62757598"],"text":"A137320 \u2192 basis A013609"},"headline":"A137320 \u2192 basis A013609","chain":null}},{"id":"l-e3055093dfba2a22","kind":"fingerprint_related","node_a_id":"n-bdd4abd52f61e954","node_b_id":"n-a12e935483e3d89d","fingerprint":"f6-02dd92f09da9263e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-02dd92f09da9263e"},"other_id":"n-a12e935483e3d89d","other":{"id":"n-a12e935483e3d89d","name":"basis_map_A013611 of basis_map_A013609 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-02dd92f09da9263e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["24","80","64"]],"card_rows":[["1"],["4","8"],["24","80","64"],["192","832","1152","512"]],"row_sums_preview":["1","12","168","2688","48384","967680","21288960","510935040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["basis A013609","basis A013611"],"edges":["e-03658933e548767a","e-ff0a190c8a29413f"],"text":"A028338 \u2192 basis A013609 \u2192 basis A013611"},"headline":"A028338 \u2192 basis A013609 \u2192 basis A013611","chain":null}}],"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:           6,           12;\n  2:          54,          180,           144;\n  3:         648,         2808,          3888,          1728;\n  4:        9720,        49896,         92016,         72576,         20736;\n  5:      174960,      1014768,       2255040,       2410560,       1244160,        248832;\n  6:     3674160,     23409648,      59533056,      77682240,      55054080,      20155392,       2985984;\n  7:    88179840,    605921472,    1709709120,    2578770432,    2253484800,    1144378368,     313528320,     35831808;\n  8:  2380855680,  17418037824,   53433203904,   90143311104,   91789334784,   57940033536,   22197805056,   4729798656,   429981696;\n  9: 71425670400, 551111402880, 1812012571008, 3345497779968, 3835399776768, 2839673023488, 1361214554112, 408267620352, 69657034752, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n+3+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3*n + 3 + 12*x)*P[n - 1, 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] = (3*n + 3 + 12*x)*v[i - 1]); 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-260e022530e5976c","steps":["basis A013611","basis A013610"],"edges":["e-10584eb473514fc3","e-58fd3bd2ee248a68"],"text":"A286718 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A286718 \u2192 basis A013611 \u2192 basis A013610","chain":null}