{"id":"n-392ebb75e65f97c1","revision":0,"first_index":0,"fingerprint":"f6-5162632e955115eb","name":"basis_map_A008277 of basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(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"],["6","4"],["108","192","48"],["3240","11760","7200","960"],["136080","913920","1034880","322560","26880"],["7348320","87151680","163296000","89510400","16934400","967680"],["484989120","9910010880","28982257920","24908083200","8036582400","1021870080","42577920"],["37829151360","1313520848640","5790091507200","7335568880640","3593682892800","763294371840","69742632960","2214051840"],["3404623622400","199297876377600","1294730467430400","2328009088204800","1626049487462400","513305778585600","77181847142400","5313724416000","132843110400"],["347271609484800","34112153390515200","321698106842112000","800412452775936000","766447126308864000","337452519282278400","74267534986444800","8319698318131200","447149909606400","9033331507200"],["39588963481267200","6507701786518732800","88164220077680947200","298190114969370624000","381206245208076288000","224951526124919193600","67978886407726694400","11018857772482560000","955139306913792000","41191991672832000","686533194547200"],["4988209398639667200","1369942848101360332800","26466927652905772032000","120121657944947295436800","201165254943558377472000","154844857861684287897600","61722626631361029734400","13577690907801590169600","1687785219805741056000","116563038436196352000","4123318366450483200","57668788341964800"],["688372897012274073600","315545770327987725926400","8648385446260983744921600","52165323733275511337779200","112840811532375690181017600","111104428456094574890188800","56834339298914065003315200","16296235939986058248192000","2724271845627963978547200","267876135351493853184000","15086270367834675609600","445664396306703974400","5305528527460761600"],["103255934551841111040000","78955279871698158888960000","3058489483224143083315200000","24339234224600078476492800000","67278978716134180733337600000","83501959649699042696724480000","53736197319795006241505280000","19535234478879556011294720000","4217681847345171604439040000","553697126681864943697920000","44136824458159262269440000","2065707532166847528960000","51728903142742425600000","530552852746076160000"],["16727461397398259988480000","21329077206290101740011520000","1164636194884869486653153280000","12159184697343845592573542400000","42592350653640511831063756800000","65884505375431219492301045760000","52544531471201451666865520640000","23736854562149293092451123200000","6437146899095138115374284800000","1083402181170513379432857600000","114617225658675217658019840000","7555482209906444489195520000","298517154256137989652480000","6417567306816537231360000","57299708096576225280000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","348","23160","2434320","365208480","73386371520","18905944337280","6047425736352000","2343513046754880000","1079054159142037478400","581082476245155082598400","361213443771404010077491200","256346786173611615389107200000","205746435539474382869965608960000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-6","-4"],["12","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4"],["0","8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":198010,"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-38283ca4d946889f"]},{"id":198011,"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-38283ca4d946889f"]},{"id":198012,"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-38283ca4d946889f"]}],"region_exact":{},"incoming_edges":[{"id":"e-38283ca4d946889f","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-392ebb75e65f97c1","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"dc281359d31b2b49653335aac39db20454f9d5db5820958df5cddb05c3a71e74","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-b4b87865f0ed6035","role":"sequence","name":"basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(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":[{"id":"l-051b19b90775d447","kind":"fingerprint_related","node_a_id":"n-a75d298d4f18bd00","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-a75d298d4f18bd00","other":{"id":"n-a75d298d4f18bd00","name":"basis_map_A008277 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["9","16","4"]],"card_rows":[["1"],["3","2"],["9","16","4"],["27","98","60","8"]],"row_sums_preview":["1","5","29","193","1449","12077","110309","1093001"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A008277"],"edges":["e-db20805112d196a7"],"text":"A013609 \u2192 basis A008277"},"headline":"A013609 \u2192 basis A008277","chain":null}},{"id":"l-213bd5791f9b71b5","kind":"fingerprint_related","node_a_id":"n-eb4206c692b7ba1c","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-eb4206c692b7ba1c","other":{"id":"n-eb4206c692b7ba1c","name":"basis_map_A008277 of basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3"],["27","18"],["243","432","108"]],"card_rows":[["3"],["27","18"],["243","432","108"],["2187","7938","4860","648"]],"row_sums_preview":["3","45","783","15633","352107","8804133","241245783","7171179561"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["basis A013610","basis A008277"],"edges":["e-284e6c48cfe1aa93","e-57babc6bb811fbca"],"text":"A120909 \u2192 basis A013610 \u2192 basis A008277"},"headline":"A120909 \u2192 basis A013610 \u2192 basis A008277","chain":null}},{"id":"l-2651936489cf2d43","kind":"fingerprint_related","node_a_id":"n-84c3b1d2f94c6d82","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-84c3b1d2f94c6d82","other":{"id":"n-84c3b1d2f94c6d82","name":"basis_map_A008277 of derivative of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["12","8"]],"card_rows":[["2"],["12","8"],["54","96","24"],["216","784","480","64"]],"row_sums_preview":["0","2","20","174","1544","14490","144924","1544326"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["D","basis A008277"],"edges":["e-9f3f778aa1812acb","e-49695d56c016a57f"],"text":"A013609 \u2192 D \u2192 basis A008277"},"headline":"A013609 \u2192 D \u2192 basis A008277","chain":null}},{"id":"l-3069821d8ba92441","kind":"fingerprint_related","node_a_id":"n-2b7e8f1d82ad0654","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-2b7e8f1d82ad0654","other":{"id":"n-2b7e8f1d82ad0654","name":"basis_map_A008277 of basis_map_A013611 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","8"],["216","384","96"]],"card_rows":[["1"],["12","8"],["216","384","96"],["4320","15680","9600","1280"]],"row_sums_preview":["1","20","696","30880","1622880","97388928","6523233024","480150967296"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013611","basis A008277"],"edges":["e-6d76bdb8087cd7e0","e-64211e3f438d055a"],"text":"A356546 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A356546 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-39449216e205f928","kind":"fingerprint_related","node_a_id":"n-cae3750b8f2fc5bd","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-cae3750b8f2fc5bd","other":{"id":"n-cae3750b8f2fc5bd","name":"basis_map_A008277 of basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["72","128","32"]],"card_rows":[["1"],["6","4"],["72","128","32"],["1080","3920","2400","320"]],"row_sums_preview":["1","10","232","7720","324576","16231488","931890432","60018870912"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013611","basis A008277"],"edges":["e-9f3d1c5f0f510549","e-f03d123bd8270b8b"],"text":"A085880 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A085880 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-3a1eb941178da379","kind":"fingerprint_related","node_a_id":"n-c9a78a8323866503","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-c9a78a8323866503","other":{"id":"n-c9a78a8323866503","name":"basis_map_A008277 of basis_map_A013609 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["36","64","16"]],"card_rows":[["1"],["6","4"],["36","64","16"],["216","784","480","64"]],"row_sums_preview":["1","10","116","1544","23184","386464","7059776","139904128"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013609","basis A008277"],"edges":["e-6abeab86de871564","e-826e6e32e99d3a96"],"text":"A130123 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A130123 \u2192 basis A013609 \u2192 basis A008277","chain":null}},{"id":"l-3e87a56ff661550c","kind":"fingerprint_related","node_a_id":"n-a67efd75592de231","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-a67efd75592de231","other":{"id":"n-a67efd75592de231","name":"basis_map_A008277 of basis_map_A013611 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","8"],["216","384","96"]],"card_rows":[["1"],["12","8"],["216","384","96"],["5184","18816","11520","1536"]],"row_sums_preview":["1","20","696","37056","2782080","278254080","35581271040","5640934440960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013611","basis A008277"],"edges":["e-413ab29272bdfecc","e-89ae27aa488fe0da"],"text":"A156992 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A156992 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-4c25e042b715e628","kind":"fingerprint_related","node_a_id":"n-5abce33cf81d1896","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-5abce33cf81d1896","other":{"id":"n-5abce33cf81d1896","name":"basis_map_A008277 of basis_map_A013610 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["9","6"],["81","144","36"]],"card_rows":[["1"],["9","6"],["81","144","36"],["729","2646","1620","216"]],"row_sums_preview":["1","15","261","5211","117369","2934711","80415261","2390393187"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A013610","basis A008277"],"edges":["e-14a3a792ab5fe5af","e-ab6865a2ee65d7b6"],"text":"A013609 \u2192 basis A013610 \u2192 basis A008277"},"headline":"A013609 \u2192 basis A013610 \u2192 basis A008277","chain":null}},{"id":"l-50ecc7c5f5477704","kind":"fingerprint_related","node_a_id":"n-4c6d389a1e69ba8f","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-4c6d389a1e69ba8f","other":{"id":"n-4c6d389a1e69ba8f","name":"basis_map_A008277 of basis_map_A013611 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["72","128","32"]],"card_rows":[["1"],["6","4"],["72","128","32"],["1296","4704","2880","384"]],"row_sums_preview":["1","10","232","9264","556416","46375680","5083038720","705116805120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013611","basis A008277"],"edges":["e-b8ab5069d7db191f","e-496e0133f5bd48c6"],"text":"A196347 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A196347 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-56c617cdba509f4f","kind":"fingerprint_related","node_a_id":"n-a8805f831bd9f0e3","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-a8805f831bd9f0e3","other":{"id":"n-a8805f831bd9f0e3","name":"basis_map_A008277 of basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["36","24"],["756","1344","336"]],"card_rows":[["1"],["36","24"],["756","1344","336"],["12096","43904","26880","3584"]],"row_sums_preview":["1","60","2436","86464","2921184","97388928","3261616512","110804069376"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013611","basis A008277"],"edges":["e-377b2cf400ddb478","e-2d69bca5c312cb8d"],"text":"A178822 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A178822 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-6c7e9b4bfd65a83f","kind":"fingerprint_related","node_a_id":"n-906c9612be68275f","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-906c9612be68275f","other":{"id":"n-906c9612be68275f","name":"basis_map_A008277 of basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["9","16","4"]],"card_rows":[["1"],["3","2"],["9","16","4"],["27","98","60","8"]],"row_sums_preview":["1","5","29","193","1449","12077","110309","1093001"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013609","basis A008277"],"edges":["e-7869781294d5f3a8","e-9f1053a900658563"],"text":"basis-map coefficients of A026729 \u2192 basis A013609 \u2192 basis A008277"},"headline":"basis-map coefficients of A026729 \u2192 basis A013609 \u2192 basis A008277","chain":null}},{"id":"l-6cc858d87d50864f","kind":"fingerprint_related","node_a_id":"n-3408a1ca2fd4b6f2","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-3408a1ca2fd4b6f2","other":{"id":"n-3408a1ca2fd4b6f2","name":"basis_map_A008277 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","8"],["144","256","64"]],"card_rows":[["1"],["12","8"],["144","256","64"],["1728","6272","3840","512"]],"row_sums_preview":["1","20","464","12352","370944","12366848","451825664","17907728384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A013611","basis A008277"],"edges":["e-e6fcabb7b5383fc4","e-903a3620f35ed697"],"text":"A038208 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A038208 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-84690fddcaeeb3a2","kind":"fingerprint_related","node_a_id":"n-a4fd53d9bf6db01c","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-a4fd53d9bf6db01c","other":{"id":"n-a4fd53d9bf6db01c","name":"basis_map_A008277 of basis_map_A013611 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["12","8"],["72","128","32"]],"card_rows":[["2"],["12","8"],["72","128","32"],["432","1568","960","128"]],"row_sums_preview":["2","20","232","3088","46368","772928","14119552","279808256"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013611","basis A008277"],"edges":["e-3deac81ab612cd2e","e-145d8cf42652d35f"],"text":"A028326 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A028326 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-a59e9bd96a1db3fa","kind":"fingerprint_related","node_a_id":"n-f1b46426574116be","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-f1b46426574116be","other":{"id":"n-f1b46426574116be","name":"basis_map_A008277 of basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["30","20"],["540","960","240"]],"card_rows":[["1"],["30","20"],["540","960","240"],["7560","27440","16800","2240"]],"row_sums_preview":["1","50","1740","54040","1622880","48694464","1482552960","46168362240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013611","basis A008277"],"edges":["e-4faaf8fb320b6fca","e-ee05e31bfcd6535a"],"text":"A178821 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A178821 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-a825fc2780f6ab0d","kind":"fingerprint_related","node_a_id":"n-8ed7c3031d730586","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-8ed7c3031d730586","other":{"id":"n-8ed7c3031d730586","name":"basis_map_A008277 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["18","12"],["324","576","144"]],"card_rows":[["1"],["18","12"],["324","576","144"],["5832","21168","12960","1728"]],"row_sums_preview":["1","30","1044","41688","1877904","93910752","5146576704","305970327936"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013611","basis A008277"],"edges":["e-fc7ebcb9132c171a","e-defbb26ad7c4a73b"],"text":"A038221 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A038221 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-b4c02122d8b64b5a","kind":"fingerprint_related","node_a_id":"n-13ed3bdd70382ee4","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-13ed3bdd70382ee4","other":{"id":"n-13ed3bdd70382ee4","name":"basis_map_A008277 of basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["18","32","8"]],"card_rows":[["1"],["3","2"],["18","32","8"],["162","588","360","48"]],"row_sums_preview":["1","5","58","1158","34776","1449240","79422480","5508725040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013609","basis A008277"],"edges":["e-456dc355a6d78eff","e-9ec4870009831d7a"],"text":"A136572 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A136572 \u2192 basis A013609 \u2192 basis A008277","chain":null}},{"id":"l-cfe568460f1b2be8","kind":"fingerprint_related","node_a_id":"n-2a90a7c6a6537016","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-2a90a7c6a6537016","other":{"id":"n-2a90a7c6a6537016","name":"basis_map_A008277 of basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-3","-2"],["9","16","4"]],"card_rows":[["1"],["-3","-2"],["9","16","4"],["-27","-98","-60","-8"]],"row_sums_preview":["1","-5","29","-193","1449","-12077","110309","-1093001"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013609","basis A008277"],"edges":["e-66f8525b021adc45","e-db82b63d1b999d44"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013609 \u2192 basis A008277"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013609 \u2192 basis A008277","chain":null}},{"id":"l-e30ac4393435c782","kind":"fingerprint_related","node_a_id":"n-19066bc55e80a423","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-19066bc55e80a423","other":{"id":"n-19066bc55e80a423","name":"basis_map_A008277 of basis_map_A013609 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-3","-2"],["9","16","4"]],"card_rows":[["1"],["-3","-2"],["9","16","4"],["-27","-98","-60","-8"]],"row_sums_preview":["1","-5","29","-193","1449","-12077","110309","-1093001"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A013609","basis A008277"],"edges":["e-13dcb978bba71940","e-b4f23fa5413158a7"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013609 \u2192 basis A008277"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013609 \u2192 basis A008277","chain":null}},{"id":"l-e961113416e6c360","kind":"fingerprint_related","node_a_id":"n-b35e74b1fec1401c","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-b35e74b1fec1401c","other":{"id":"n-b35e74b1fec1401c","name":"basis_map_A008277 of basis_map_A013609 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["9","16","4"],["27","98","60","8"],["81","544","616","192","16"]],"card_rows":[["9","16","4"],["27","98","60","8"],["81","544","616","192","16"],["243","2882","5400","2960","560","32"]],"row_sums_preview":["29","193","1449","12077","110309","1093001","11653905","132827925"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013609","basis A008277"],"edges":["e-7bee75fc05394bb8","e-9ff0dc0fa016f11d"],"text":"basis-map coefficients of A112227 \u2192 basis A013609 \u2192 basis A008277"},"headline":"basis-map coefficients of A112227 \u2192 basis A013609 \u2192 basis A008277","chain":null}},{"id":"l-f1b7cb29b816aa89","kind":"fingerprint_related","node_a_id":"n-469b0008ce00f6df","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-469b0008ce00f6df","other":{"id":"n-469b0008ce00f6df","name":"basis_map_A008277 of basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["24","16"],["360","640","160"]],"card_rows":[["1"],["24","16"],["360","640","160"],["4320","15680","9600","1280"]],"row_sums_preview":["1","40","1160","30880","811440","21641984","593021184","16788495360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013611","basis A008277"],"edges":["e-159425c642d412bb","e-98d37e4f2a05dba7"],"text":"A178820 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A178820 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-f547d1f70bab4327","kind":"fingerprint_related","node_a_id":"n-20d6880016d8135a","node_b_id":"n-392ebb75e65f97c1","fingerprint":"f6-5162632e955115eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5162632e955115eb"},"other_id":"n-20d6880016d8135a","other":{"id":"n-20d6880016d8135a","name":"basis_map_A008277 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-5162632e955115eb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3"],["9","6"],["27","48","12"]],"card_rows":[["3"],["9","6"],["27","48","12"],["81","294","180","24"]],"row_sums_preview":["3","15","87","579","4347","36231","330927","3279003"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["basis A008277"],"edges":["e-1fc5789fa458cf8e"],"text":"A120909 \u2192 basis A008277"},"headline":"A120909 \u2192 basis A008277","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,                 4;\n  2:             108,               192,                 48;\n  3:            3240,             11760,               7200,                960;\n  4:          136080,            913920,            1034880,             322560,              26880;\n  5:         7348320,          87151680,          163296000,           89510400,           16934400,             967680;\n  6:       484989120,        9910010880,        28982257920,        24908083200,         8036582400,         1021870080,          42577920;\n  7:     37829151360,     1313520848640,      5790091507200,      7335568880640,      3593682892800,       763294371840,       69742632960,       2214051840;\n  8:   3404623622400,   199297876377600,   1294730467430400,   2328009088204800,   1626049487462400,    513305778585600,    77181847142400,    5313724416000,    132843110400;\n  9: 347271609484800, 34112153390515200, 321698106842112000, 800412452775936000, 766447126308864000, 337452519282278400, 74267534986444800, 8319698318131200, 447149909606400, 9033331507200;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12*n-6+(8*n-4)*x)*P(n-1,x) + (8*n-4)*x*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[(12*n - 6 + (8*n - 4)*x)*P[n - 1, x] + (8*n - 4)*x*D[P[n - 1, x], 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] = (12*n - 6 + (8*n - 4)*x)*v[i - 1] + (8*n - 4)*x*deriv(v[i - 1], 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-c7809537965f400d","steps":["basis A013611","basis A008277"],"edges":["e-20dcbad557ba463b","e-38283ca4d946889f"],"text":"A085881 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A085881 \u2192 basis A013611 \u2192 basis A008277","chain":null}
